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

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

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. हत्योः परस्परं यच्छेषं गुणकारभागहारकयोः ।
   तेन हृतौ निश्छेदौ तावेव परस्परं हत्योः । —BrSpSi. XVIII. 9.

226 BRAHMAGUPTA AS AN ALGEBRAIST
der. The residue (and the divisor corresponding to the
smaller remainder) being mutually divided, the last resi-
due should be multiplied by such an optional integer
that the product being added(in case the number of quo-
tients of the mutual division is even) or subtracted (in
case the number of quotients is odd) by the difference
of the remainders (will be exactly divisible by the last
but one remainder. Place the quotients of the mutual
division successively one below the other in a column;
below them the optional multiplier and underneath it
the quotient just obtained). Any number below . : .
the penultimate) is multiplied by the one just above
it and then added by that just below it. Divide the
last number (obtained by doing so repeatedly) by the
divisor corresponding to the smaller remainder; then
multiply the residue by the divisor corresponding to
the greater remainder and add the greater remainder.
(The result will be) the number corresponding to the
two divisors.¹
There is an alternative rendering of this passage also as
follows : .
Divide the divisor corresponding to the greater remain-
der by the divisor corresponding to the smaller remain-
der. The residue (and the divisor corresponding to
the smaller remainder) being mutually divided (until
the remainder becomes zero), the last quotient should
be multiplied by an optional integer and then added
(in case the number of quotients of the mutual division
is even) or subtracted (in case the number of quotients
is odd) by the difference of the remainders. (Place the
other quotients of mutual division successively one
below the other in a column; below them the result just
obtained and underneath it the optional integer). Any

1. अधिकाग्रभागहारं छिन्यादूनाग्रभागहारेण ।
   शेषपरस्परभक्तं मतिगुणमग्रान्तरे क्षिप्तम् ॥
   अध उपरि गुणितमन्त्ययुगूनाच्छेद भाजिते शेषम् ।
   अधिकाग्रच्छेदगुणं द्विच्छेदाग्रमधिकाग्रयुतम् । —Ārya. II. 32-33

PRELIMINARY OPERATIONS 227
number below (i.e. the penultimate) is multiplied by
the one just above it and then added by that just
below it. Divide the last number (obtained
by doing so repeatedly) by the divisor corres-
ponding to the smaller remainder; then multiply the
residue ty the divisor corresponding to the greater
remainder and add the greater remainder. (The result
will be) the number corresponding to the two divisors.
Āryabhaṭa’s problem may be enunciated thus :
To find a number (N) which being divided by two
given numbers (a, b) will leave two given remainders
(R₁, R₂).
This gives :
N=ax+R₁=by+R₂
(where R₁ is a greater remainder and R₂ lesser remainder, and a
is the divisor corresponding to greater remainder and b the
divisor corresponding to the lesser remainder.)
Denoting as before by c the difference between R₁, and
R₂, we get
(i) by=ax+c, if R₁>R₂
(ii) ax=by+c, if R₂>R₁
the equation being so written as to keep c always positive.
Hence the problem now reduces to making either
(ax+c)/b or (by+c)/a
according as R₁>R₂ or R₂>R₁, a positive integer. So Āryabhaṭa
says : Divide the divisor corresponding to the greater remainder
etc.”
Now we shall proceed with the details of the operation as
proposed by Datta and Singh in his History of Hindu Mathema-
tics, Part II. Algebra :
Suppose R₁>R₂; then the equation to be solved will be
ax+c=by ...(i)
a, b being prime to each other.

228 BRAHMAGUPTA AS AN ALGEBRAIST
Let
b) a (q
bq
───
r₁) b (q₁
r₁q₁
───
r₂) r₁ (q₂
r₂q₂
───
r₃
...
──────
rₘ₋₁) rₘ₋₂ (qₘ₋₁
rₘ₋₁ qₘ₋₁
───────
rₘ) rₘ₋₁ (qₘ
rₘqₘ
─────
rₘ₊₁
Then we get (when a < b, we shall have q = 0, r₁ = a)
a = bq + r₁
b = r₁q₁ + r₂
r₁ = r₂q₂ + r₃
r₂ = r₃q₃ + r₄
... ... ...
rₘ₋₂ = rₘ₋₁ qₘ₋₁ + rₘ
rₘ₋₁ = rₘqₘ + rₘ₊₁
Now, substituting the value of a in the given equation (1),
we get
by = (bq + r₁)x + c
Therefore
y = qx + y₁
where
by₁ = r₁x + c
In other words, since a = bq + r₁, on putting
y = qx + y₁ (ii)
the given equation (i) reduces to
by₁ = r₁x + c (iii)
Again, since b = r₁q₁ + r₂

PRELIMINARY OPERATIONS 229
putting similarly x=q₁y₁+x₁
the equation (iii) can be further reduced to
r₁x₁=r₂y₁—c (iv)
and so on.
Writing down the successive values and reduced equations
in columns, we have
(1) y=qx+y₁ (I.1) by₁=r₁x+c
(2) x=q₁y₁+x₁ (I.2) r₁x₁=r₂y₁—c
(3) y₁=q₂x₁+y₂ (I.3) r₂y₂=r₃x₁+c
(4) x₁=q₃y₂+x₂ (I.4) r₃x₂=r₄y₂—c
(5) y₂=q₄x₂+y₃ (I.5) r₄y₃=r₅x₂+c
(6) x₂=q₅y₃+x₃ (I.6) r₅x₃=r₆y₃—c
......... .........
(2n-1) yₙ₋₁=q₂ₙ₋₂ xₙ₋₁+yₙ (I. 2n-1) r₂ₙ₋₂ yₙ=r₂ₙ₋₁ xₙ₋₁+c
(2n) xₙ₋₁=q₂ₙ₋₁ yₙ+xₙ (I. 2n) r₂ₙ₋₁ xₙ=r₂ₙ yₙ—c
(2n+1) yₙ=q₂ₙ xₙ+yₙ₊₁ (I. 2n+1) r₂ₙ yₙ₊₁=r₂ₙ₊₁ xₙ+c
Now the mutual division can be continued either (i) to the
finish or (ii) so as to get a certain number of quotients and
then stopped. In either csse the number of quotients found, negle-
cting the first one (q), as is usual with Āryabhaṭa, may be even
or odd.
Case (i) First suppose that the mutual division is continued
until the zero remainder is obtained. Since a, b are prime to
each other, the last one remainder is unity.
Subcase (i.1.). Let the number of quotients be even. We
then have
r₂ₙ=1, r₂ₙ₋₁=0, q₂ₙ=r₂ₙ₋₁
The equations (1,2n) and (I.2n+1), therefore become
yₙ=q₂ₙ xₙ+c
and
yₙ₊₁=c
respectively. Giving an arbitrary integral value (t) to xₙ
we get an integral value of yₙ. From that we can find the value
of xₙ₋₁ by the equation (2n). Procceding backwards step by
step we ultimately find the values of x and y in positive integers.
So that the equation (I) is solved.
Subcase (i. 2) : If the number of quotients be odd, we shall
have
r₂ₙ₋₁=1, r₂ₙ=0, q₂ₙ₋₁=r₂ₙ₋₂.

230 BRAHMAGUPTA AS AN ALGEBRAIST
The equations (2n+1) and (I. 2n+1) will then be absent and
the equations (I. 2n-1) and (I. 2n) will be reduced respectively
to
xₙ₋₁ = q₂ₙ₋₁ y - c
and xₙ = - c
Giving an arbitrary integral value (t') to yₙ we get an in-
tegral value of xₙ₋₁. Then proceeding backwards as before we
calculate the values of x and y.
Case (ii) : Next suppose that the mutual division is stopped
after having obtained an even or odd number of quotients.
Subcase (ii.1) : If the number of quotients obtained be
even the reduced form of the original equation is
r₂ y + 1 = r₂ₙ + 1 xₙ + c
or yₙ ₊ ₁ = (r₂ₙ ₊ ₁ xₙ + c) / r₂
Giving a suitable integral value (t) to xₙ as will make
yₙ₊₁ = (r₂ₙ₊₁ t + c) / r₂ₙ = an integral number,
we get an integral value for yₙ by (2n+1). The values of
x and y can then be calculated by proceeding as before.
Subcase (ii.2) : If the number of quotients be odd the reduc-
ed form of the quotient is
r₂ₙ₋₁ xₙ = r₂ₙ yₙ - c
or xₙ = (r₂ₙ yₙ - c) / r₂ₙ₋₁
Putting yₙ = t', where t' is an integer, such that
xₙ = (r₂ₙ t' - c) / r₂ₙ₋₁ = a whole number,
we get an integral value of xₙ ₋ ₁ by (2n). Whence can be
calculated the values of x and y in integers.
If x = α and y = β be the least integral solution of ax + c =
by, we shall have
aα + c = bβ
Therefore a(bm + α) + c = b (am + β),
m being any integer. Therefore, in general,
x = bm + α
But we have calculated before that

BHASKARA I AND KUṬṬAKA OPERATION 231
x = q₁y₁ + x₁ ;
∴ q₁y₁ + x₁ = bm + α
Thus it is found that the minimum value α of x is equal to
the remainder left on dividing its calculated value by b. whence
we can calculate the minimum value of N (=aα+R₁). This will
explain the rationale of the operations described in the latter
portion of the rule of Āryabhaṭa I.
Bhāskara I and Kuṭṭaka Operation
In Chapter I of the Mahābhāskarīya, Bhāskara I has descri-
bed the preliminary operation to be performed on the divisor
and dividend of a pulveriser. We shall quote it from the edition
of K.S, Shukla :
The divisor (which is "the number of civil days in a
yuga) and the dividend (which is "the revolution num-
ber of the desired planet") become prime to each other
on being divided by the (last non-zero) residue of
the mutual division of the number of civil days in a
yuga and the revolution number of the desired planet.
The operations of the pulveriser should be performed
on them (i. e. on the abraded divisor and abraded
dividend). So has been said.¹
An indeterminate equation of the first degree of the
type
(ax — c) / a = y
(with x and y unknown) is known in Hindu mathematics by
the name of "pulveriser"—kuṭṭakāra). In this equation, a is
called the "dividend" (bhājya), b the "divisor" (bhāgahāra), c
the interpolator (kṣepa), x the "multiplier" (guṇakāra), and y
the "quotient" (labdha).
In the pulveriser contemplated in the above stanza :
a = revolution number of a planet.-
b = civil days in a yuga,
c = residue of the revolutions of the planet (Śeṣa)

1. भूदिनेष्टगणावन्योन्य भक्तशेषेण भाजितौ
   हारभाज्यौ दृढौ स्यातां कुट्टाकारं तयोर्विदुः । — MBh. I. 41

232 BRAHMAGUPTA AS AN ALGEBRAIST
x=ahargaṇa ,
and y=complete revolutions performed by the planet.
The text says that as a preliminary operation to the solu-
tion of this pulveriser, a and b. i.e., civil days in yuga and revo-
lution-number of the planet, should be made prime to each other
by dividing them out by their greatest common factor, That is
to say, in solving a pulveriser, one should always make use of
abraded divisor and abraded dividend.
The interpolator, i.e., the residue, should also be divided
out by the same factor. (This instruction is not given in the
text, but it is implied that the residue should be computed for
the abraded dividend and abraded divisor).
Set down the dividend above and the divisor (hāra)
below that. Divide them mutually and write down
the quotients (labdha) of division one below the other
(in the form of a chain). (When an even number of
quotients is obtained) think out by what number the
(last) remainder be multiplied so that the product
being diminished by the (given) residue be exactly
divisible (by the divisor corresponding to that remain-
der). Put down the chosen number called mati below
the chain and then the new quotient underneath it.
Then by the chosen number multiply the number which
stands just above it, and to the product add the quoti-
ent (written below the chosen number). (Replace the
upper number by the resulting sum and cancel the
number below). Proceed afterwards also in the same
way (until only two numbers remain). Divide the
upper number (called the "multiplier") by the divisor
by the usual process and the lower one (called the
"quotient") by the dividend : the remainders (thus
obtained) will respectively be the ahargaṇa and the
revolutions etc. or what one wants to know.¹
We shall illustrate the operation by taking a problem from
the Laghu-Bhāskarīya (VIII. 17) :
The sum, the difference, and the product increased by
one, of the residues of the revolution of Saturn and
Mars—each is a perfect square. Taking the equations

BHĀSKARA I AND KUṬṬAKA OPERATIONS 233
furnished by the above and applying the method of
such quadratics, obtain the (simplest) solution by the
substitution of 2, 3 etc. successively in the general solu-
tion). Then calculate the ahargaṇa and the revolu-
tions performed by Saturn and Mars in that time toge-
ther with the number of solar years elapsed.¹
Let x and y denote the residues of the revolution of Mars
and Saturn respectively. Then we have to find out two numbers
x and y such that each of the expressions x+y, x-y and xy+1
may be a perfect square.
Let x+y=4P² and x-y=4Q², so that
x=2P²+2Q²
y=2P²-2Q²
and therefore xy+1=(2P²-1)²+4(P²-Q⁴)
Hence the condition that xy+1 be a perfect square is that
P²=Q⁴. Substituting these values, we have
x=2(Q⁴+Q²)
y=2(Q⁴-Q²)
where Q may possess any of the values 2, 3, 4,......but not 1. (We
neglect the case when x or y is zero).

1. भाज्यं न्यसेदुपरि हारमधश्च तस्य ।
   खण्ड्यात्परस्परमथो विनिधाय लब्धम् ।
   केनाऽऽहतोऽ

234 BRAHMAGUPTA AS AN ALGEBRAIST
Putting Q=2, we get x=40 and y=24, which is the least
solution.
Assuming now that the residues of the revolution (maṇḍa-
laja-śeṣa) of Saturn and Mars are 24 and 40 respectively, we have
to obtain the ahargaṇa (which means the number of mean civil
days elapsed since the beginning of Kaliyuga, or, in fact, any
epoch).
The revolution-number of Saturn is 146564, and the
number of civil days in a yuga is 1,577,917,500. In the present
problem, these are respectively the dividend and the divisor.
Their H.C.F. is 4, so that dividing them out by 4 we get 36641
and 394,479,375 as the abraded dividend and abraded divisor
respectively. We have, therefore, to solve the pulveriser
36641x–24
────────── = y
394479375
where x and y denote the ahargaṇa and the revolutions respecti-
vely made by Saturn.
Mutually dividing 36641 and 394479375, we get
36641) 394479375 (10766
394477006
─────────
2369) 36641 (15
35535
─────
1106) 2369 (2
157) 1106 (7
1099
7) 157 (22
154
3) 7 (2
6
──
1×27–24=3)3(1
3
──
0
We have chosen here the number 27 as the optional
number (mati). In fact, mati may be chosen at any stage after
an even number of quotients are obtained.

BHĀSKARA I AND KUṬṬAKA OPERATIONS 235
Writing down the quotients one below the other as pres-
cribed in the rule, we get the chain
10766
15
2
7
22
2
(mati) 27
1
Reducing the chain, we successively get
10766 10766 10766 10766 10766 10766 3108044439
(multiplier)
15 15 15 15 15 288689 288689
(quotient)
2 2 2 2 18665 18665
7 7 7 8714 8714
22 22 1237 1237
2 55 55
(mati)
27 27
1
(it would be seen in this reduction of chain that mati
or 27 × 2 plus 1 is 55; 55 × 22 plus 27 is 1237; 1237 × 7
plus 55 is 8714; 8714 × 2 plus 1237 is 18665; 18665 × 15
plus 8714 is 288689; and finally 288689 × 10766 plus
18665 is 3108044439 which is the multiplier).
Dividing 3108044439 by 394479375, and 288689 by 36641,
we obtain 346688814 and 32202 respectively as remainders, (This
division is performed only when the multiplier and quotient are
greater than the divisor and dividend respectively). These are
the minimum values of x and y satisfying the above equation.
Therefore, the required ahargaṇa = 346688814. and the
revolutions performed by Saturn = 32202.
To obtain the ahargaṇa and the revolutions of Mars, one
has to solve the equation :
191402 z — 40
───────────── = w
131493125

236 BRAHMAGUPTA AS AN ALGEBRAIST
where z and w denote the ahargaṇa and the revolutions per-
formed by Mars respectively.
The general solution of this equation is
z = 131493125 s + 118076020
w = 191402 s + 171872
where s = 0, 1, 2, 3......When s = 0, we have the least solution.
Brahmagupta’s Rules Concerning
Indeterminate Analysis of the First Degree
For the solution of Āryabhaṭa’s problem, Brahmagupta
gives the following rule :
What remains when the divisor corresponding to the
greater remainder is divided by the divisor correspond-
ing to the smaller remainder—that (and the latter
divisor) are mutually divided and the quotients are
severally set down one below the other. The last residue
(of the reciprocal division after an even number of quo-
tients has been obtained) is multiplied by such an optio-
nal integer that the product being added with the differ-
ence of the (given) remainders will be exactly divisible
(by the divisor corresponding to that residue). That
optional multiplier and then the (new) quotient just
obtained should be set down (underneath the listed
quotients). Now, proceeding from the lower-most
number (in the column), the penultimate is multiplied
by the number just above it and then added by the
number just below it. The final value thus obtained
(by repeating the above process) is divided by the
divisor corresponding to the smaller remainder. The
residue being multiplied by the divisor corresponding
to the greater remainder and added to the greater
remainder will be the number in view.¹

1. अधिकाग्रभागहारादूनाग्रच्छेद भाजिताच्छेषम् ।
   यत् तत् परस्परहृतं लब्धमधोधः पृथक् स्थाप्यम् ॥
   शेषं तदिष्टगुणितं यथाऽनयोरन्तरेण संयुक्तम् ।
   शुध्यति गुणकः स्थाप्यो लब्धं चान्त्यादुपान्त्यगुणः ॥
   स्वोर्ध्वोऽन्त्य युतोऽग्रान्तो हीनाग्रच्छेदभाजितः शेषम् ।
   अधिकाग्रच्छेदहत मग्रिकाप्रयुतं भवत्यग्रम् ॥ —BrSpSi. XVIII, 3-5

BRAHMAGUPTA'S RULES OF ANALYSIS 237.
Brahmagupta further observes :
Such is the process when the quotients (of mutual
division) are even in number. But if they be odd,
what has been stated before as negative should be
made as positive, or as positive should be made
negative.¹
Regarding the direction for dividing the divisor corres-
ponding to the greater number by the divisor corresponding to
the smaller remainder, Pṛthūdaka Svāmī (860A.D.) observes that
it is not absolute, rather optional; so that the process may be
conducted in the same way by starting with the division of the
divisor corresponding to the smaller remainder by the divisor
corresponding to the greater remainder. But in this case of in-
version of the process, he continues, the difference of the remain-
ders, must be negative.
That is to say, the equation
by=ax+c
can be solved by transforming it first to the form
ax=by-c
so that we shall have to start with the division of b by a.
For the details of the "Theory of the pulveriser" as applied
to the problems in Astronomy, the reader is referred to the writ-
ings of Bhaṭṭa Govind, translated by K.S. Shukla, and given as
an Appendix to the edition of the Laghu-Bhāskarīya. For the
rationale of the rules in relation to kuṭṭaka or the pulveriser
operation, one may also refer to the chapters by Datta and Singh
in the History of Hindu Mathematics: Algebra.
Solution of by=ax ± 1.
This simple indeterminate equation has a special use in
astronomical calculations and therefore, Indian algebraists have
paid special attention to it. In fact, this equation is solved
exactly in the same way as the equation by=ax ± c; it is a parti-

1. एवं समेषु विषमेष्वृणं धनं धनमृणं यदुक्तं तत् ।
   ऋणधनयोर्व्यस्तत्वं गुण्य प्रक्षेपयोः कार्यम् ॥ —BrSpSi. XVIII. 13.

238- BRAHMAGUPTA AS AN ALGEBRAIST
cular case only of the more general latter equation. Of course, there
is a little justification also for treating it separately, since both
the types of equations represent two different physical conditions
of the astronomical problems. In the case of by=ax±c, the
conditions are such that the value of either y or x, more particu-
larly of the latter, has to be found and the rules for solution
formulated with that objective. But in the case of the equation
by=ax±1, the physical conditions require the values of both
y and x.
The equation by=ax±1 is usually known by the name
sthira-kuṭṭaka, literally meaning the 'constant pulveriser' Pṛthū-
daka Svāmī also names it as dṛḍha-kuṭṭaka meaning firm-pulveri-
ser. Later on this term dṛḍha-was confined to another sense,
equivlent to nicched (having no divisor) or nirapavarta (irreduci-
ble). The origin of the name sthira-kuṭṭaka or constant pulveriser
has been explained by Pṛthūdaka Svāmī as being due to the fact
that the interpolator (±1) is here invariable.
For the solution of this equation, we shall quote Bhāskara
I's rule and the rule by Brahmagupta. Bhāskara I writes in this
connection as follows :
The method of the pulveriser is applied also after
subtracting unity. The multiplier and quotient are
respectively the numbers above and underneath. Multi-
plying those quantities by the desired number divide
by the reduced divisor and dividend; the residues are
in this case known to be the (elapsed) days and (resid-
ues of) revolutions respectively¹.
The pulveriser
ax - c
-------- = y ... (1)
b
may be written as
aX - 1
-------- = Y ... (2)
b
where x=cX and y=cY. If X=α, Y=β is a solution of (2), then
x=cα, y=cβ will be a solution of (1). Hence the above rule.

1. रूपमेकमपास्यापि कुट्टाकारः प्रसाध्यते ।
   गुणकारोऽथ लब्धं च राशी स्यातामुपर्यधः ॥
   —MBh. I. 45

BRAHMAGUPTA'S RULES OF ANALYSIS 239
Brahmagupta's Rule in this connection is as follows :
Solution of by=ax-1 :
Divide them (i.e., the abraded coefficient of the multi-
plier and the divisor) mutually and set down the quo-
tients one below the other. The last residue (or the
reciprocal division after an even number of quotients
has been obtained) is multiplied by an optional integer
such that the product being diminished by unity will
be exactly divisible (by the divisor corresponding to
that residue). The (optional) multiplier and then this
quotient should be set down (underneath the listed
quotients). Now proceeding from the lower most
term to the uppermost, by the penultimate multiply
the term just above it and then add the lowermost
number. (The uppermost number thus calculated being
divided by the reduced divisor, the residue (is the quan-
tity required. This is the method of the constant pul-
veriser¹.
Solution of
by+ax=±c
Indian algebraists usually transformed this equation as
by=-ax+c, so that it appeared as a particular case of by=ax+c,
in which a was negative. Brahmagupta has been the first
person to solve this equation, but the rule given by him is
obscure :
The reversal of the negative and positive should be
made of the multiplier and interpolator.²
Pṛthudaka Svāmī has tried to explain it, but he too is not
very clear. He says :

1. हृतयोः परस्परं यच्छ्रेषं गुणकार भागहारकयोः ।
   तेन हृतौ निश्छेदौ तावेव परस्परं हृतयोः ॥
   लब्धमधोऽधः स्थाप्यं तथेष्ट गुणकारसङ्गुणं शेषम् ।
   शुद्धस्यति यथैकहीनं गुणकः स्थाप्यः फलं चान्त्यम् ॥
   अश्रान्तमुपान्त्येन स्वोर्ध्वो गुणितोऽन्त्य संयुतो भक्तम् ।
   निःशेषभागहारेणैव स्थिरकुट्टकः शेषम् ॥ —BrSpSi. XVIII. 9-11
2. एवं समेषु विषमेष्वृणं धनं धनमृणं यदुक्तं तत् ।
   ऋणधनयोर्व्यस्तत्वं गुण्यप्रक्षेपयोः कार्यम् ॥ BrSpSi XVIII. 13

240 BRAHMAGUPTA AS AN ALGEBRAIST
If the multiplier be negative, it must be made positive;
and the additive must be made negative : and then the
method of the pulveriser should be employed.
Pṛthūdaka Svāmī, however, does not indicate how to derive
the solution of the equation.
by = -ax+c ...(1)
from that of the equation
by = ax-c ...(2)
The method, however, seems to have been this :
Let x=α, y=β be the minimum solution of (2). Then
we get
bβ = a α-c
or b(a-β) = -a(a-b)+c
Hence x=a-b, y=a-β is the minimum solution of (1). This
rule is very clearly indicated by Bhāskara II and others.
We shall give two examples from Bhāskara II (Bījagaṇita)
to illustrate the rule :
Example I.
13y = -60x + 3
By the method described before, we find that the minimum
solution of
13y = 60x+3
is x=11, y=51. Subtracting these values from their respective
abraders, namely 13 and 60, we get 2 and 9. Then by the
maxim : "In the case of the dividend and divisor being of differ-
ent signs, the results from the operation of division should be
known to be so", making the quotient negative we get the solu-
tion of
13y = -60x+3
as x=2, y=-9. Subtracting these values again from their respec-
tive abraders (13, 60), we get the solution of
13y = -60x-3
as x=11, y=-51.
Example II.
11y = 18x+10

LINEAR EQUATIONS 241
Proceeding as before, we find the minimum solution of
11y=18x+10
to be x=8, y=14. These will also be the values of x and y in
the case of the negative divisor but the quotient for the reasons
stated before should be made negative. So the solution of
-11y=18x+10
is x=8, y=-14. Subtracting these (i.e., their numerical values)
from their respective abraders, we get the solution of
-11y=18x-10
as x=3, y=-4,
"When the divisor is positive or negative the numeri-
cal values of the quotient and multiplier remain the
same : when either the divisor or the dividend is
negative, the quotient must always be known to be
negative"¹.
One Linear Equation in
More Than Two Unknowns
Whenever a linear equation involves more than two unkno-
wn's the Indian algebraists used to assume arbitrary values for all
the unknowns except two and then to apply the method of kuṭ-
ṭaka or "pulveriser". In this connection, Brahmagupta says :
The method of the pulveriser (should be employed if
there be present many unknowns (in any equation)²,

1. Bhāskara II gives the following rule :
   "Those(the multiplier and quotient)cbtained for a positive divi-
   dend being treated in the same manner give the results corres-
   ponding to a negative dividend."
   The treatment alluded to in this rule is that of subtraction
   from the respective abraders. He has further elaborated it
   thus :
   The multiplier and quotient should be determined by
   taking the dividend, divisor and interpolator as positive. They
   will be the quantities for the additive interpolator.
   Subtracting them from their respective abraders, the quantities
   for a negative interpolator are found. If the dividend or divi-
   sor, be negative, the quotient should be stated as negative, the
   quotient should be stated as negative.
   —Bījagaṇita
2. आद्याद्वर्णादन्यान् वर्णान् प्रोह्याद्यमानमाद्यहृतम् ।
   सदृशच्छेदावसकृद् द्वौ व्यस्तौ कुट्टको बहुषु ॥ — BrSpSi. XVIII. 51