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

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

242 BRAHMAGUPTA AS AN ALGEBRAIST
We shall take up one of the problems posed by
Brahmagupta concerning astronomy and leading to
the equation :¹
197x - 1644 y - z = 6302.
Hence
1644 y + z + 6302
x = -------------------
197
The commentator assumes z = 131. Then
1644 y + 6433
x = --------------- ;
197
hence by the usual method of the pulveriser
x = 41; y = 1.
General Problem of Remainders
A certain type of simultaneous indeterminate equations of
the first degree arise out of the general problem of remainders
which may thus be stated : To find a number N which being
severally divided by a₁, a₂, a₃.......aₙ , leaves as remainders r₁, r₂,
r₃.........rₙ respectively.
While dealing with such a case, we shall have the following
series of equations :
N = a₁x₁ + r₁ = a₂x₂ + r₂ = a₃x₃ + r₃ = ...... = aₙ xₙ + r .
We have reasons to believe that the method of solution of
these equations was known to Āryabhaṭa I. In the translation of
the verse in the Āryabhaṭīya, II. 32-33 (the translation of which
we have already given), the term dvicchedāgram should be
translated as "the result will be the remainder corresponding to
the product of the two divisors", instead of "the result will be the
number corresponding to the two divisors." (the last line of the
translation). This explanation is in fact given by Bhāskara I, the
direct disciple and earliest commentator of Āryabhaṭa I. Such a
rule is clearly stated by Brahmagupta².

1: अंशकशेषेण युतात् लिप्ताशेषात्तदन्तरादथवा ।
भानोर्भ'दिने द्युगणं यः कथयति कुट्टकज्ञः सः ॥ —BrSpSi. XVIII. 55
2: स्वोच्छेऽन्ययुतोऽग्रान्तो हीनाग्रच्छेदभाजितः शेषम् ।
अधिकाग्रच्छेदहतमधिकाग्रयुतं भवत्यग्रम् ॥ —BrSpSi. XVIII. 5

GENERAL PROBLEM OF REMAINDERS 243
The rationale of this method is not difficult. I shall quote it
from the book of Datta and Singh: Starting with the considera-
tion of the first two divisors, we have
N = a₁x₁ + r₁ = a₂x₂ + r₂.
By the method described before, we can find the minimum
value α of x₁ satisfying this equation. Then the minimum value
of N will be a₁α + r₁. Hence the general value of N will be given
by
N = a₁ (a₂t + α) + r₁
= a₁a₂ t + a₁α + r₁
where t is an integer. Thus a₁α + r₁ is the remainder left on
dividing N by a₁ a₂ as stated by Āryabhaṭa I and Brahmagupta.
Now taking into consideration the third condition, we have
N = a₁a₂t + a₁αr₁ = a₃x₃ + r₃
which can be solved in the same way as before. Proceeding in
this way successively, we shall ultimately arrive at a value of
N satisfying all the conditions ;
Pṛthūdaka Svāmī remarks :
Wherever the reduction of two divisors by a common
measure is possible, there 'the product of the divisors'
should be understood as equivalent to the product of
the divisor corresponding to the greater remainder and
quotient of the divisor corresponding to the smaller
remainder as reduced (i.e. divided) by the common mea-
sure.¹ When one divisor is exactly divisible by the
other, then the greater remainder is the (required)
remainder and the divisor corresponding to the greater
remainder is taken as 'the product of the divisors'.
(The truth of) this may be investigated by an intelligent
mathematician by taking several symbols.
As an illustration we shall take up a problem quoted by
Bhāskara II in his Bījagaṇita, and which in its solution follows
the method of Āryabhaṭa 1. Pṛthūdaka Svāmī while commenting
on serveral verses from Brahmagupta (BrSpSi. XVIII. 3-6)

1. i.e., if p be the L.C.M. of a, and a2, the general value of N satisfying
   the above two conditions will be
   N = pt + a₁a + r,
   instead of N = a₁a₂t + a₁a + r₁.

244 BRAHMAGUPTA AS AN ALGEBRAIST
observes that such problems were very popular amongst the an-
cient Indian mathematicians.
Problem : To find a number N which leaves remainders 5,
4,3,2 when divided by 6,5,4,3 respectively.
That is to solve the equations :
N = 6x + 5 = 5y + 4 = 4z + 3 = 3w + 2.
We have since N = 6x + 5 = 5y + 4,
x = (5y - 1) / 6
But x must be integral, so y = 6t + 5, x = 5t + 4
Hence N = 30t + 29
Again N = 30t + 29 = 4z + 3
Therefore, t = (2z - 13) / 15
Since t must be integral, we must have z = 15s + 14;
hence t = 2s + 1. Therefore
N = 60s + 59.
The last condition is identically satisfied. The method given
here is the one followed by Pṛthūdaka Svāmī.
Thus when N = 60s + 59 = 6x + 5
x = (60s + 54) / 6 = 10s + 9 ...(1)
Again, when N = 60s + 59 = 5y + 4,
y = (60s + 55) / 5 = 12s + 11
Again when N = 60s + 59 = 4z + 3
z = (60s + 56) / 4 = 15s + 14
Lastly, when N = 60s + 59 = 3w + 2.
w = (60s + 57) / 3 = 20s + 19,
Varga Prakṛti or Kṛti Prakṛti or Square-Nature
The word varga-prakṛti (literally meaning 'square-nature')
has been given by Indian algebraists to the indeterminate quadra-
tic equation
Nx² ± c = y²

VARGA PRAKṚTI 245
Here in this equation the absolute number c should be
rūpa (or unity). which means the equation
Nx²±1=y²
or it may be any absolute number. The most fundamental
equation of this class has been regarded as
Nx²+1=y²
where N is a non-square integer.
This branch of mathematics has originated from the number
which is the prakṛti of the square of yāvat, etc. (the unknown
x etc.), and therefore, it is called varga-prakṛti. The quantity
N of the above equation is known as Prakṛti. Brahmagupta uses
the term GUṆAKA (multiplier) for the same purpose¹.
This term guṇaka together with its variation guṇa appears
occasionally also in the writings of later authors. For example,
Śrīpati (Siddhānta-śekhara. XIV. 32) employs the term guṇaka
where as Bhāskara II and Nārāyaṇa use the term guṇa in their
Bījagaṇitas.
In this connection, we would now like to quote from
Pṛthūdaka Svāmī (863 A.D.) from his commentary on the
Brāhmasphuṭasiddhānta :
Here are stated for ordinary use the terms which
are well known to people.. The number whose square,
multiplied by an optional multiplier and then increased
or decreased by another optional number, becomes
capable of yielding a square-root, is designated by the
term the "lesser root" kaniṣṭha pada or the "first root"
ādya-mūla). The root which results, after those opera-
tions have been performed is called by the name the
"greater root" (jyeṣṭha pada) or the "second root"
(anya-mūla). If there be a number multiplying both
these roots, it is called the "augmenter" (udvartaka);
and on the contrary, if there be a number dividing the
roots, it is called the "abridger" (apavartaka),
Thus in the equation
Nx²±c=y²,

1. मूलं द्विधेष्ट वर्गाद् गुणक गुणादिष्टयुत विहीनाच्च ।
   आद्यवधो गुणकगुणः सहान्त्यघातेन कृतमन्त्यम् ॥ —BrSpSi. XVIII. 64
2. BrSpSi. XVIII. 64 (Com.)

246 BRAHMAGUPTA AS AN ALGEBRAIST
x is known as the lesser root, y is the greater root, N is the multi-
plier (guṇaka) and c is interpolator or kṣepaka. Bhāskara II has
used the word "hrasvamūla" for kaniṣṭha pada or ādya-mūla lite-
rally meaning "lesser root". The earlier terms, the "first root"
(ādyamūla) for the value of x and the "second root" or the "last
root" antya-mūla for the value of y are quite free from ambiguity
Their use is found in the algebra of Brahmagupta. The later
terms appears in the works of his commentator Pṛthūdaka Svāmī.
Brahmagupta uses the term kṣepa, prakṣepa or prakṣepaka
in the sense of "interpolator." Again, when negative, the inter-
polator is sometimes distinguished as the "subtractive" or
śodhaka and the positive interpolator is then called "the addi-
tive."
Lemmas of Brahmagupta
Prior to our giving the general solution of the Square-nature
or Varga-Prakṛti, it would be better to give two Lemmas esta-
blished by Brahmagupta. We have the following in the Brāhma-
sphuṭasiddhānta:
Of the square of the optional number multiplied by the
guṇaka and increased or decreased by an other optional
number, iṣṭa, (extract) the square root. (Proceed) twice.
The product of the first roots multiplied by the guṇaka
together with the product of the second roots will
give a (fresh) second root; the sum of their cross-pro-
ducts will be a (fresh) first root. The (corresponding)
interpolator will be equal to the product of the (previ-
ous) interpolators.¹
There is a little difficulty in ascertaining the real sense of the
rule given in these lines since the word dvidhā (twice) has two
implications. Firstly, it may mean that the earlier operations of
finding roots are made on two optional numbers with two optio-
nal interpolators, and with the results thus obtained the subse-

1. मूलं द्विधेष्टवर्गाद् गुणक गुणादिष्टयुत विहीनाच्च ।
   आद्यवधो गुणकगुणः सहान्त्यघातेन कृतमन्त्यम् ॥
   वज्रवधैक्यं प्रथमं प्रक्षेपः क्षेपवध तुल्यः ।
   प्रक्षेपशोधकहृते मूले प्रक्षेपके रूपे ॥ —BrSpSi. XVIII 64-65

LEMMAS OF BRAHMAGUPTA 247
quent operations of their composition are performed. Secondly,
it may also mean that the earlier operations are made with
one optionally chosen number and one interpolator, and the
subsequent ones are carried out after the repeated statement of
those roots for the second time. It is also implied that in the
composition of the quadratic roots, their products may be added
together or subtracted from each other.
In other words, if x=a, y=β be a solution of the equation :
Nx² + k = y²,
and x=a', y=β' be a solution of
Nx² + k' = y²,
then according to the above
x = aβ' ± a'β, y = ββ' ± Naa'
is a solution of the equation
Nx² + kk' = y².
In other words, if
Na² + k = β²
Na'² + k' = β'²
then
· N(aβ' ± a'β)² + kk' = (ββ' ± Na')² (I)
In particular, taking a=a', β=β' and k=k', Brahmagupta
finds from a solution x=a, y=β of the equation
Nx² + k² = y²
a solution x=2aβ, y=β² + Na² of the equation
Nx² + k = y²
That is, if
Na² + k = β²
then
N(2aβ)² + k² = (β² + Na²)² (II)
This result will be hereafter called Brahmagupta's Corol-
lary.
Thus Brahmagupta's First Lemma says that if two solutions
of the equation (of the Square-nature) Nx² + 1 = y² are known,
then any number of other solutions can be found. For example
if two solutions of the Square-nature are (a, b) and also (a', b'),
then two other solutions will be :
x = ab' ± a'b, y = bb' ± Naa'.

248 BRAHMAGUPTA AS AN ALGEBRAIST
We can compose this solution with the previous ones, and
get another solution, and thus proceed on to innumerable solu-
tions. From Brahmagupta's Corollary to First Lemma we get
another set of solutions. If (a, b) be solution of the Square-
nature, then another solution of it is
x=2ab, and y=b²+Na²
Thus even if we have only one solution, we can get the
other solution also (since N is known), and thus we can get any
number of solutions one after the other by this Principle of
Composition.
Brahmagupta's Lemmas have been described by Bhāskara
II (1150 A.D.) in the following words :
Set down successively the lesser root (hrasva), greater
root (jyeṣṭha) and interpolator (kṣepaka); and below
them should be set down in order the same or an
another (set of similar quantities). From them by the
Principle of Composition (Bhāvanā) can be obtained
numerous roots. Therefore the Principle of Composi-
tion will be explained here. (Find), the two cross-pro-
ducts (vajrābhyāsa) of the two lesser and the two greater
roots; their sum is a lesser root. Add the product of
the two lesser roots multiplied by the prakṛti to the
product of the two greater roots, the sum will be a
greater root. In that (equation) the interpolator will
be the product of the two previous interpolators. Again
the difference of the two cross-products is a lesser root.
Subtract the product of the two lesser roots multiplied
by the prakṛti from the product of the two greater
roots; (the difference) will be greater root. Here also
the interpolator is the product of the two (previous)
interpolators.¹

1. ह्रस्वज्येष्ठ क्षेपकान् न्यस्य तेषां तानन्यान् वाऽधो निवेश्य क्रमेण ।
   साध्यान्येभ्यो भावनाभिर्बहूनि मूलान्येषां भावना प्रोच्यतेऽतः ।।
   वज्राभ्यासौ ज्येष्ठलघ्वोस्तदैक्यं ह्रस्वं लघ्वोराहतिश्च प्रकृत्या ।
   क्षुण्णा ज्येष्ठाभ्यास युग् ज्येष्ठमूलं तत्राभ्यासः क्षेपयोः क्षेपकः स्यात् ।।
   ह्रस्वं वज्राभ्यासयोरन्तरं वा लघ्वोर्घातो यः प्रकृत्या विनिघ्नः ।
   घातो यश्च ज्येष्ठयोस्तद्वियोगो ज्येष्ठं क्षेपोत्रापि च क्षेपघातः ।।
   Bhāskara II, Bījaganita, VargaPrakṛti, 2-4

PRINCIPLE OF COMPOSITION 249
Principle of Composition
The above results have been technically known amongst
Indian algebraists as Bhāvanā (demonstrated or proved, hence
theorem or lemma). The word bhāvanā also means "composition
or combination" in algebra. Bhāvanā may be of two types :
Samāsa Bhāvanā (or addition Lemma, or additive composition)
and Antara Bhāvanā (or subtraction Lemma or subtractive
composition). Whenever, again, the bhāvanā is made with two
equal sets of roots and interpolators, it is technically named as
Tulya Bhāvanā (or composition of equals), and when with two
unequal sets of values then it is known as Atulya Bhāvanā (or
composition of unequals).
Proof of Brahmagupta's Lemmas
It is significant to be indicated that Brahmagupta's Lemmas
were rediscovered by Euler in 1764 and by Lagrange in 1768,
and a considerable importance was attached to them. Kṛṣṇa,
(1580 A.D.) the commentator on the Bījagaṇita of Bhāskara II
gives the following proof of Brahmagupta's Lemmas :
Let (α,β) and (α',β') be the two solutions of the equation
nx² + k = y².
we have
Nα² + k = β²
Nα'² + k' = β'²
Multiplying the first equation by β'², we get
Nα²β'² + kβ'² = β²β'²
Now, substituting the value of factor β²' of the interpolator
from the second equation, we get
Nα² β'² + k (Nα'² + k') = β²β'²
or N(α²β'² + Nkα'² + kk' = β²β'²
Again, substituting the value of k from the first equation
in the second term of the left-hand side expression, we have
Nα²β'² + Nα'²(β² - Nα²) + kk' = β²β'²
or N(α²β'² + α'²β²) + kk' = β²β'² + N²α²α'²
Adding ±2Nαβα'β' to both sides, we get
N(αβ' ± α'β)² + kk' = (ββ' ± Nαα')²

250 BRAHMAGUPTA AS AN ALGEBRAIST
Brahmagupta's Corollary also follows at once from the
above by putting α'=α, β'=β and k'=k.
N (2αβ)²+k²=(β²±Nα²)²
Thus the roots are x=2αβ and y=β²±Nα² which is the
Corollary.
It would be seen that modern historians of mathematics are
incorrect when they say that Fermat (1657) was the first to
state that the equation Nx²+1=y², where N is a non-square
integer has an unlimited number of solutions in integers. For
this assertion, history takes us to the early Seventh Century A.D.
when Brahmagupta wrote his classical treatise, the Brāhmasphu-
tasiddhānta, and gave the well known two Lemmas and the
Corollary to the first Lemma.
Second Lemma of Brahmagupta
In the Brāhmasphuṭa siddhānta, we find another important
Lemma by Brahmagupta stated as follows :
On dividing the two roots (of a square- Nature) by the
square-root of its additive or subtractive, the roots for
interpolator unity (will be found).¹
This Lemma when expressed in the modern language of
algebra would mean that if x=α, y=β be a solution of the
equation.
Nx²+k²=y²
then x=α/k, y=β/k is a solution of the equation
Nx²+1=y².
This rule, at another place, has been re-enunciated as
follows :
If the interpolator is that divided by a square then
the roots will be those multiplied by its square-
root.²

1. प्रक्षेपशोधक हृते मूले प्रक्षेपके रूपे ।
   —BrSpSi. XVIII. 65
2. वर्गाच्छिन्ने क्षेपे तत्पदगुणिते तदा मूले ।
   —BrSpSi. XVIII. 70

SECOND LEMMA OF BRAHMAGUPTA 251
This rule may be expressed in terms of symbols as follows.
Suppose the Varga-prakṛti (Square-nature) to be
Nx² ± p²d = y²,
so that its interpolator (kṣepa) p²d is exactly divisible by the
square p². Then, putting therein u = x/p, v = y/p, we derive the
equation
Nu² ± d = v²
whose interpolator is equal to that of the original Square-nature
divided by p². It is clear that the roots of the original equation
are p times those of the derived equation.
Rational Solution
Indian algebraists have usually suggested the following
method to obtain a first solution of Nx² + 1 = y² :
Take an arbitrary small rational number, α, such that
its square multiplied by the guṇaka N and increased
or diminished by a suitably chosen rational number k
will be an exact square.
In other words, we shall have to obtain empirically a rela-
tion of the form
Nα² ± k = β²
where α, k, and β are rational numbers. Let us call this
relation as the Auxiliary Equation. Then by Brahmagupta's Coro-
llary, we get from it the relation
N(2αβ)² + k² = (β² + Nα²)²,
or N(2α β / k)² + 1 = ((β² + Nα²) / k)²
Hence, one rational solution of the equation Nx² + 1 = y² is
given by
x = 2αβ / k , y = (β² + Nα²) / k
Work on the rational solution of the Square-nature has been
also done by Śrīpati. In fact, his solution, given in 1039 A.D.
is of historical significance. He derives the rational solution
without the aid of the "auxiliary equation." He gives the follo-
wing rule :

252 BRAHMAGUPTA AS AN ALGEBRAIST
Unity is the lesser root. Its square multiplied by the
prakṛti is increased or decreased by the prakṛti com-
bined with an (optional) number whose square-root
will be the greater root. From them will be obtained
two roots by the Principle of Composition¹
Thus if m² be the rational number optionally chosen, one
shall have the identity :
N.1²+(m²—N)=m²,
or N.1²—(N—m²)=m²
Then by applying Brahmagupta's Corollary we get
N(2m)²+(m² ∼ N)²=(m²+N)²
∴ N (2m / (m² ∼ N))² + 1 = ((m² + N) / (m² ∼ N))²
Hence
x = 2m / (m² ∼ N), y = (m² + N) / (m² ∼ N)
where m is any rational number, is a solution of the equation
Nx² + 1 = y².
This rational solution of the varga-prakṛti which was used
by Śrīpati in 1039 A.D. was rediscovered in Europe by Broun-
cker in 1657.
We shall close this discussion by taking an illustration
from Bhāskara II :
Problem : Tell me, O mathematician, what is that square
which multiplied by 8 becomes, together with
unity, a square; and what square multiplied by
11 and increased by unity, becomes a square.
This means that we have to solve the equations :
8x² + 1 = y² ......(i)
11x² + 1 = y² ......(ii)
In the second example, let us assume 1 as the lesser root.
Following the method of Śrīpati, let us multiply its square by
the prakṛti (here in eq. ii, prakṛti is 11), then let us subtract 2
(an optional number) and then extracting the square-roots we

1. Śrīpati, Siddhānta-śekhara XIV. 33

RATIONAL SOLUTION 253
get the greater root as 3. Hence the statement for the com-
position is
m=11 l=1 g=3 i=-2
l=1 g=3 i=-2
Here m=multiplier (guṇaka or prakṛti), l=lesser root
(kaniṣṭha-mūla), g=greater root (jyeṣṭha-mūla) and i=interpola-
tor (kṣepa).
Here we have set down successively the lesser root, greater
root and interpolator, and below them again set down the same
(See Brahmagupta's Lemmas described by Bhāskara II). Now
proceeding as before we obtain the roots for the additive 4 :
l=6, g=20, (for) i=4.
Then by the rule :
"If the interpolator (of a varga-prakṛti or Square-nature)
divided by the square of an optional number be the
interpolator (of another Square-nature), then the two
roots (of the former) divided by that optional number
will be the roots (of the other). Or, if the interpolator
be multiplied, their roots should be multiplied."¹
are found the roots for the additive unity
l=3, g=10 (for) i=1.
Whence by the Principle of Composition of Equals, we get
the lesser and greater roots : l=60, g=199 (for) i= 1. In this
way an infinite number of roots can be deduced.
Alternative method:—Bhāskara II has given another method
for finding the two roots for the additive unity :
Or divide twice an optional number by the difference
between the square of that optional number and the
prakṛti. This (quotient) will be the lesser root (of a
Square-nature) when unity is the additive. From that
(follows) the greater root.²

1. इष्टवर्गहृतः क्षेपः क्षेपः स्यादिष्टभाजिते ।
   मूले ते स्तोऽथवा क्षेपः क्षुण्णः क्षुण्णे तदा पदे ॥
   Bījagaṇita II. 5.
2. Siddhānta-śekhara, XIV. 32.

254 BRAHMAGUPTA AS AN ALGEBRAIST
Let us solve the first example 8x²＋1＝y². We assume the
optional number to be 3. Its square is 9; the prakṛti of multiplier
is 8, their difference is 9–8＝1. Dividing by this twice the optional
number (2×3, i.e. 6), namely 6, we get the lesser root for the addi-
tive unity as 6. Whence proceeding as before, we get the greater
to be 17. Thus here x＝6 and y＝17.
Let us use this method for the equation 11x²＋1＝y². Let
the optional number be 3. Its square is 9: multiplier or prakṛti
is 11; the difference is 11–9＝2; dividing by this twice the
optional number (2×3), namely 6, we get 6/2＝3, which is the
lesser root. Consequently the greater root would be 10. Thus
for this equation x＝3 and y＝10.
Solution in Positive Integers
The Indian algebraists usually aimed at obtaining solutions
of the varga-prakṛti or Square-nature in positive integers or
abhinna. The tentative methods of Brahmagupta and Śrīpati
always did not furnish solutions in positive integers. These auth-
ors, however, discovered that if the interpolator of auxiliary equa-
tion in the tentative method be ±1, ±2 or ±4, an integral solu-
tion of the equation Nx²＋1＝y² can always be found. Thus
Śrīpati says :
If 1, 2 or 4 be the additive or subtractive (of the auxi-
liary equation), the lesser and greater roots will be
integral (abhinna)¹.
(i) If k＝±1, then the auxiliary equation will be
Nα²±1＝β
where α and β are intergers. Then by Brahmagupta' Corollary
we get
x＝2αβ and y＝β²＋Nα²
as the required first solution in positive integers of the equation
Nx²＋1＝y²

1. इष्टवर्गं प्रकृत्योर्यद्विवरं तेन वा भजेत् ।
   द्विघ्नमिष्टं कनिष्ठं तत् पदं स्यादेक संयुतौ ।
   ततो ज्येष्ठमिहानन्त्यं भावनाभिस्तथेष्टतः ।।
   Bījagaṇita, Varga-Prakṛti, 5-6

Here is the complete line-by-line transcription of the page into clean Markdown, preserving all text, mathematical expressions (without LaTeX math mode), and Devanagari Sanskrit:

SOLUTION IN POSITIVE INTEGERS 255
(ii) Let k=±2; then the auxiliary equation is
Nα²±2=β²
By Brahmagupta's Corollary, we have
N(2αβ)²+4=(β²+Nα²)²
or N (αβ)²+1=( (β²+Nα²)/2 )²
Hence the required first solution is
x=αβ, y=½(β²+Nα²)
Since Na²=β² ∓ 2,
we have ½ (β²+Nα²)=β² ∓ 1=a whole number.
(iii) Now suppose k=+4: so that
Nα²+4=β²
With an auxiliary equation like this, the first integral solution
of the equation Nx²+1=y² is
x=½αβ
y=½(β²-2);
if α is even; or
x=½

256 BRAHMAGUPTA AS AN ALGEBRAIST
Substituting the value of N in the right-hand side expres-
sion from (i), we have
N.( αβ / 2 )² + 1 = ( (β²-2) / 2 )² (iii)
Composing (ii) and (iii),
N { α/2 (β²-1) }² + 1 = { β/2 (β²-3) }²
Hence x= ½ αβ, y= ½ (β²--2);
and x= ½α(β²-1), y= ½β(β²-3);
are solutions of Nx²+1=y².
If β be even, the first values of (x,y) are integral. If β be
odd, the second values are integral.
(iv) Finally, suppose k=-4; the auxiliary equation is
Nα²-4 = β²
Then the required first solution in positive integers of
Nx²+1=y² is
x= ½αβ(β²+3) (β²+1)
y=(β²+2) { ½(β²+3) (β²+1)-1 }.
Brahmagupta says :
In the case of 4 as subtractive, the square of the second
is increased by three and by unity; half the product of
these sums and that as diminished by unity (are
obtained). The latter multiplied by the first sum less
unity is the (required) second root; the former multi-
plied by the product of the (old) roots will be the first
root corresponding to the (new) second root.¹
The rationale of this solution, as given by Datta and Singh
is as follows :
Nα²-4=β² (i)
N(α/2)²-1=(β/2)²
Hence by Brahmagupta's Corollary, we get
N ( αβ / 2 )² + 1 = ( β²/4 + N α²/4 )²

1. चतुरूनेऽन्यपद कृती त्र्येकयुते वधदलं पृथग्व्येकम् ।
   व्येकाद्वाहतमन्त्यं पदवध गुणमाद्यमान्त्यपदम् ॥ BrSpSi. XVIII. 68

CAKRAVĀLA OR CYCLIC METHOD 257
= {½(β² + 2)}² (ii)
Again applying the Corollary, we get
N {½αβ(β² + 2)}² + 1 = {½(β⁴ + 4β² + 2)}² (iii)
Now by the Lemma we obtain from (ii) and (iii)
N {½αβ(β² + 3) (β² + 1)}² + 1
= [(β² + 2){½(β² + 3) (β² + 1) − 1}]²
Hence x = ½αβ(β² + 3) (β² + 1),
y = (β² + 2){½(β² + 3) (β² + 1) − 1}
is a solution of Nx² + 1 = y²
This can be proved without difficulty that these values of x
and y are integral. Since if β is even, β² + 2 is also even. And
hence the above values of x and y are integral. On the other
hand, if β is odd, β² is also odd; under these conditions β² + 1 and
β² + 3 are even. In this also, therefore, the above values must be
integral.
Putting p = αβ, q = β² + 2, we can write the above solution
in the form
x = ½p (q² − 1).
y = ½q (q² − 3).
This was the form in which the solution was found by
Euler.
Cakravāla or Cyclic Method
We have shown in the preceding articles that the most
fundamental step in Brahmagupta's method for the general
solution in positive integers of the equation
Nx² + 1 = y²
where N is a non-square integer, is to form an auxiliary equation
of the kind
Na² + k = b²
where a and b are positive integers and k = ±1, ±2 or ±4. From
this auxiliary equation, by the Principle of Composition, applied
repeatedly whenever necessary, one can derive, as we have alrea-
dy shown above, one positive integral solution of the original
Varga-prakṛti or Square-Nature. And thence again, by means
of the same principle, an infinite number of other solutions in
integers can be obtained. How to form an auxiliary equation of

258 BRAHMAGUPTA AS AN ALGEBRAIST
this type was a problem, write Datta and Singh, which could
not be solved completely nor satisfactorily by Brahmagupta. In
fact, Brahmagupta had to depend on trial. Success in this direc-
tion was, however, remarkably attained by Bhāskara II. He evol-
ved a simple and elegant method which assisted in deriving an
auxiliary equation having the required interpolator ±1, ±2, or ±
4, simultaneously with its two integral roots, from another auxi-
liary equation empirically formed with any simple integral value
of the interpolator, positive or negative. This method has been
technically known as Cakravāla or the cyclic method. This is
so called because it proceeds as in a circle, the same set of opera-
tions being applied again and again in a continuous round. For
the details of this method, our reader is requested to consult
the Algebra of Bhāskara II and the narrative on this method as
given by Datta and Singh under the title "Cyclic Method" in
their History of Hindu Mathematics: Algebra, 1962 Edition,
pp. 161-72.
Solution of Indeterminate
Quadratic Equation
It is remarkable to see that Brahmagupta was the first
algebraist in the history of mathematics to find a general solu-
tion of the indeterminate quadratic equation
Nx²±c=y²
in positive integers. We have the following verse in the
Brāhmasphuṭasiddhānta in this connection:
From two roots (of a Square-nature or varga-prakṛti)
with any given additive or subtractive, by making
(combination) with the roots for the additive unity
other first and second roots (of the equation having)
the given additive or subtractive (can be found).¹
Let us take the following two equations:
a₁k=an+b; and b₁k=bn+Na
From them we get : by eliminating n
a₁b−ab₁=1

1. रूप प्रक्षेपपदे पृथगिष्टक्षेपशोध्यमूलाभ्याम् ।
   कृत्वाऽऽन्त्याथपदे ये प्रक्षेपे तेने ॥ —BrSpSi. XVIII. 66

SOLUTION OF INDETERMINATE 259
Hence b₁ = (a₁b - 1) / a = a whole number.
Now n² - N = ((a₁k - b)² - Na²) / a²
= (a₁²k² - 2bka₁ + k) / a²
= k(a₁²k - 2ba₁ + 1) / a²
Therefore (k / a²)(a₁²k - 2ba₁ + 1) is a whole number.
Since a, k have no common factor, it follows that
(a₁²k - 2ba₁ + 1) / a² = (n² - N) / k = k₁ = an integer.
Also k₁ = (n² - N) / k = (a₁²k - 2ba₁ + 1) / a²
= (a₁²(b² - Na²) - 2ba₁ + 1) / a²
= ((a₁b - 1) / a)² - Na₁².
Thus having known a single solution in positive integers
of the equation Nx² ± c = y², says, Brahmagupta, an infinite
number of other integral solutions can be obtained by making
use of the integral solutions of Nx² + 1 = y². If (p, q) be a
solution of the former equation found empirically and if (α, β)
be an integral solution of the latter, then by the principle of
Composition
x = pβ ± qα; y = qβ ± Npα
will be a solution of the former. Repeating the operations, we
can easily deduce as many solutions as we like.
FORM Mn²x² ± c = y² :
In this connection, Brahmagupta says :
If the remainder is that divided by a square, the first
root is that divided by its root¹.
This seems to mean that if we have the equation
Mn²x² ± c = y² (i)
such that the multiplier (i.e. the coefficient of x²) is divisible

1. वर्गेच्छिन्ने गुणके प्रथमं तन्मूल भाजितं भवति । —BrSpSi. XVIII.70

260 BRAHMAGUPTA AS AN ALGEBRAIST
by n², then we are justified in saying that if we put nx=u, the
equation (i) becomes Mu²±c=y² (ii),
and clearly the first root of (i) is equal to the first root of (ii)
divided by n. The corresponding second root will be the same
for both the equations.
FORM a²x²±c=y² :
We find Brahmagupta giving the following rule in this
connection : This is a solution of a particular form of a
varga-prakṛti or Square-nature.
If the multiplier be a square, the interpolator divided
by an optional number and then increased and
decreased by it, is halved. The former (of these
results) is the second root; and the other divided by
the square-root of the multiplier is the first root.¹
Thus the solutions of the equation
a²x²±c=y²
are :
x = 1/(2a) (±c/m - m)
y = 1/2 (±c/m + m)
where m is an arbitrary number.
Bhāskara II and Nārāyaṇa have also given the same
solutions as proposed by Brahmagupta.
Rational Geometrical Figures
In the days of the Taittirīya Saṁhitā and the Śatapatha
Brāhmaṇa, Indian mathematicians got familiarity with the
solution of such equations
x²+y²=z²
and the results were arrived geometrically on the basis of the
law of rectangle as propounded by Baudhāyana in the Śulba Sūtras
and which goes by his name. The reader is referred to the Chapter
on Baudhāyana, the first Geometer in the author's "Founders
of Sciences in Ancient India". Baudhāyana (c. 800 B.C.) gave a

1. वर्गे गुणके क्षेपः केनचिदुद्धृतयुतोनितो दलितः ।
   प्रथमोऽन्यन्मूलमन्त्यो गुणकारपदोद्धृतः प्रथमः ॥ —BrSpSi. XVIII. 69