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

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

, is there a space or a matra?
Let's compare त in कोटिः or तद्दलं: त has a left loop.
In आयतचतुरस्रस्य: आ य त च तु र स्र स्य.
No, it's आयतचतुरस्रस्य. Wait, let me re-examine: आयताचतुरस्रस्य?
Wait, look at त: the right side of त goes down, then there is a च. It's आयतचतुरस्रस्य.
Wait, what about चेष्टस्येष्टाधिका?
Let's look at चे ष्टस्येष्टाधिका:
चे has a space after it?
Wait, look at the footnote line 1:
1. इष्टस्य भुजस्य कृतेर्भक्तो नेष्टेन तद्दलं कोटिः ।
Line 2:
आयतचतुरस्रस्य चे ष्टस्येष्टाधिका कर्णः ॥ —BrSpSi. XII. 35
Wait, look at चे ष्टस्येष्टाधिका:
There is चे, then a space, then ष्टस्येष्टाधिका.
Wait, could it be चेष्टस्येष्टाधिका with just a broken space? Yes, in De

262 BRAHMAGUPTA AS AN ALGEBRAIST
The sum of the squares of two unequal numbers is the
side; their product multiplied by two is the altitude,
and twice the difference of the squares of those two
unequal numbers is the base of an isosceles triangle.¹
Thus if m,n be two integers such that m is not equal to n,
the sides of all rational isosceles triangles with integral sides are
given by
m² + n², m² + n², 2(m² - n²)
and the altitude of the triangle is 2mn.
This method was also followed by Mahāvīra and other
Indian mathematicians. In fact, their solutions are based on
the juxtaposition of two rational right triangles, equal so that
they have a common leg. It is remarkably a powerful device, for
every rational triangle or quadrilateral may be formed by the
juxtaposition of two or four rational right triangles.
Isosceles Triangles with a Given Altitude
Here we have a rule given by Brahmagupta for finding out
all rational isosceles triangles possessing the same altitude :
The (given) altitude is the producer (karaṇī). Its
square divided by an optional number is increased and
diminished by that optional number. The smaller is the
base and half the greater is the side.²
Thus if m be any rational number then for a given definite
altitude a, the sides of the rational isosceles triangles are
½ (a²/m + m) each and the base is (a² - m²)/m . We shall illustrate it
by an example taken from the commentary of Pṛthūdaka Svāmī
The given altitude is 8; let us take any rational number m = 4
then the two equal sides of the isosceles are given by ½ ((8² + 4²)/4) =
10 each and the base is (8² - 4²)/4 = 12. Thus the three sides of the

1. कृति युतिर सदृशराश्योर्बाहुर्घातो द्विसंगुणो लम्बः ।
   कृत्यन्तरमसदृशयोर्द्विगुणं द्विसमत्रिभुज भूमिः ॥ —BrSpSi. XII. 33
2. करणी लम्बस्तत्कृतिरिष्टहतेष्टोन संयुताऽल्पा भूः ।
   अधिको द्विहृतो बाहुः संक्षेप्यो यद्वधो वर्गः ॥ —BrSpSi. XII. 37.

ISOSCELES TRIANGLES 263
rational isosceles triangle with altitude 8 are (10,10, 12).
Rational Scalene Triangles: Brahmagupta lays down the
following rule in the case of rational scalene triangle :
The square of an optional number is divided twice
by two arbitrary numbers; the moieties of the sums
of the quotients and (respective) optional numbers are
the sides of a scalene triangle; the sum of the moieties
of the differences is the base.¹
In other words, if m, p, q are any rational numbers, then
the sides of a rational scalene triangle are :
½ ( m²/p + p ), ½ ( m²/q + q ),
½ ( m²/p - p ) + ½ ( m²/q - q )
Here the altitude (m), area and segments of the base of
this triangle are all rational.
Thus putting m=12, p=6, and q=8 in Brahmagupta's gene-
ral equation, Pṛthūdaka Svāmī derives a scalene triangle with
sides (13,15) and (14) altitude (12), area (84 and the segments of
the base (5) which are all integral numbers.
½ ( m²/p + p ) = ½ ( 12²/6 + 6 ) = 15;
½ ( m²/q + q ) = ½ ( 12²/8 + 8 ) = 13
B' A C' A' B' C'
B H C H B C
Fig. 19 Fig. 20

1. इष्टद्वयेन भक्तो द्विधेष्ट वर्ग फलेष्टयोगार्धे ।
   विषमत्रिभुजस्य भुजाविष्टोनफलार्धयोगो भूः ॥
   —BrSpSi. XII, 34.

264 BRAHMAGUPTA AS AN ALGEBRAIST
Thus the two sides of the rational scalene triangle are 15
and 13. The base is .
½ ( 12²/6 - 6 ) + ½ ( 12²/8 - 8 ) = 9 + 5 = 14
The altitude is m = 12; area is equal to (base × altitude) / 2
= (14 × 12) / 2 = and the segments are √(13² - 12²) = 5 and
√(15² - 12²) = 9. Thus they are all integers.
Rational Isosceles Trapeziums
Brahmagupta has given us a method of obtaining such
isosceles trapeziums whose sides, diagonals, altitude, segments
and area are all rational numbers. His rule is as follows :
The diagonals of the rec-
tangle (generated) are the flank
sides of an isosceles trapezium;
the square of its side is divided
by an optional number and
then lessened by that optional
number and divided by two;
(the result) increased by the
upright is the base and lessened
by it is the face.¹
Fig. 21
Here in the figure, we have the isosceles trapezium ABCD
of which C D is the base and A B is known as the fase. Accord-
ing to Brahmagupta's rule, we have ( p being the optional
number).
CD = ½ ( (4m²n² - p) / p ) + ( m² - n² ) (base)
AB = ½ [ 4m²n²/p - p ] - [ m² - n² ] (face)
DH = ( m² - n² ) (upright)

1. आयतकर्णौ बाहू भुजकृतिरिष्टेन भाजितेष्टोना ।
   द्विहृता कोट्यधिका भूर्मुखमूना दिसमचतुरस्रे ॥ —BrSpSi. XII. 36

RATIONAL TRAPEZIUMS 265
AD = BC = m²+n² (the sides of the trapezium)
HC = base-upright = ½ [ 4m²n²/p - p ] (segment)
AC = BD = [ 4m²n²/p + p ] (diagonal)
AH = 2 mn (altitude)
ABCD = mn [ 4m²n²/p - p ] (area)
By chosing the values of m n and p suitably, the values
of all the dimensions of the isosceles trapezium can be made
integral. Pṛthūdaka Svāmī starts with the rectangle (5, 12, 13)
and suitably takes p as 6; then he calculates out the dimensions
of the trapezium : flank sides (AD and BC) = 13, base =14,
and base = 4, altitude (AH) = 12, segments of base (DH and
HC) = 5, and 9, diagonals (AC and BD) = 15, area ABCD =
108. All these values are integers.
In this example, the rectangle chosen is (5, 12, 13) which is
AA' DH, where AD = m² + n² = 13
and DH = m² - n² = 5
whence by adding the two we have
2m² = 18
This gives the value of m = 3, and hence n = 2. Pṛthū-
daka Svāmī has taken the value of p = 6 by choice. Putting
these values of m, n and p, the values for the dimensions of the
isosceles trapezium follow from the expressions given by Brahma-
gupta.
CD = ½ ( (4.3².2² / 6) - 6 ) + ( 3²-2² ) = 9+5 = 14 (base)
Face = 9-5=4
Sides AD = BC = 3²+2² = 13
and so on for the other dimensions.
Rational Trapeziums With
Three Equal Sides
This problem is very much the same as one of the rational
isosceles tpapezium with the only difference that in this case
one of the parallel sides is also equal to the slant sides. We

266 BRAHMAGUPTA AS AN ALGEBRAIST
have the following solution of this problem from Brahma-
gupta :
The square of the diagonal (of a generated rectangle)
gives three equal sides; the fourth (is obtained) by
subtracting the square of the upright from thrice the
square of the side (of that rectangle). If greater, it
is the base; if less, it is the face.¹
As before, the rectangle generated from m, n is given by
(m²–n², 2mn, m²+n²), that is these are the three sides of
the right triangle, which correspond to the two sides and the
diagonal of the rectangle generated by them. Let us suppose,
we have a trapezium ABCD whose sides AB, BC and AD are
equal, then
AB = BC = AD = (m²+n²)²
CD = 3(2mn)² – (m²–n²)² = 14 m²n² – m⁴ – n⁴
or CD = 3(m²–n²)² – (2mn)² = 3m⁴+3n⁴ – 10 m²n².
Pṛthudaka Svāmī has taken an illustration, where m=2,
n=1 and he deduces two rational trapeziums with three equal
sides (25, 25, 25, 39) and (25, 25, 25, 11).
The segment (CH), altitude (AH), diagonals (AC, BD)
and area of this trapezium are also rational, and given by :
CH (segment) = 6 m²n²—m⁴ – n⁴
AH (altitude) = 4 mn (m²—n²)
AC = BD (diagonals) = 4 mn (m²+n²)
ABCD (area) = 32m³n³ (m²—n²).
Rational Inscribed Quadrilaterals
We find in the Brāhmasphuṭasiddhānta a remarkable
proposition formulated by Brahmagupta :
To find all quadrilaterals which will be inscribable
within circles and whose sides, diagonals, perpendi-
culars, segments (of sides and diagonals by perpendi-
culars from vertices as also of diagonals by their
intersection), areas, and also the diameters of the

1. कर्णकृतित्रिसम भुजास्त्रयश्चतुर्थो विशोष्य कोटि कृतिम् ।
   बाहुकृतेस्त्रिगुणाया यद्यधिको भूर्मुखं हीनः ॥ —BrSpSi. XII. 37

RATIONAL INSCRIBED QUADRILATERALS 267
circumscribed circles will be expressible in integers.
Such quadrilaterals we shall call as Brahmagupta
Quadrilaterals.
The solution of this formidable problem has been given by
Brahmagupta as follows :
The upright and bases of two right-angled triangles
being reciprocally multiplied by the diagonals of the
other will give the sides of a quadrilateral of unequal
sides : ( of these ) the greatest is the base, the least
is the face, and the other two sides are the two
flanks.¹
Taking Brahmagupta's integral solution, the sides of the
two right triangles of reference are given by :
(1) m²-n², 2 mn, m²+n²;
(ii) p²-q², 2 pq, p²+q²;
where m, n, p, q are integers. Then the sides of the Brah-
magupta Quadrilateral are.
(m²-n²)(p²+q²), (p²-q²)(m²+n²),
2mn(p²+q²), 2pq(m²+n²) (Arrangement A)
Pṛthūdaka Svāmī has-
illustrated the rational inscribed
quadrilateral by taking an
example of the right angle
triangles.
(i) (3,4,5) (m²-n²=3,
m²+n²=5, whence
m=2, n=1)
(ii) (5,12,13)(p²-q²=5,
p²+q²=13, whence
p=3, q=2)
Fig. 22
Substituting these values in the above equations, we get the
sides of the quardilateral as ( 39, 25, 52 and 60).²

1. जात्यद्वय कोटिभुजाः परकर्णगुणा भुजाश्चतुर्विषमे ।
   अधिको भूर्मुखंहीनो बाहुद्वितयं भुजावन्यौ ॥ —BrSpSi. XII. 38
   2 The diagonals of this quadrilateral are given by Bhāskara II as 56
   (=3.12+4.5) and 63 (=4.12+3.5). (Cont. on page 268)

270 BRAHMAGUPTA AS AN ALGEBRAIST
Put in other words, this means that one has to solve the
following equations :
(i) 5x—25 = y²
(ii) 10x—100 = y²
(iii) 83x—7635 = y²
Pṛthūdaka Svāmī, the commentator on the Brāhmasphuṭa-
siddhānta proceeds to solve these equations as follows :
(1.1) Suppose y = 10; then x = 125. Or put y = 5; then
x = 10.
(2.1) Suppose y = 10; then x = 20.
(3.1) Assume y = 1; then x = 92.
He then remarks that by virtue of the multiplicity of
suppositions there will be an infinitude of solutions in every
case, But no method has been given either by Brahmagupta or
his commentator to obtain the general solution.
Double Equations of the First Degree
Perhaps we have the earliest reference of the simultaneous
indeterminate quadratic equations of the type
x ± a = u²
x ± b = v²
in the Bhakaśālī Manuscript (Folio 59, recto).
Brahmagupta gives the solution of such simultaneous inde-
terminate quadratic equations of a general case as follows :
The difference of the two numbers by the addition
or subtration of which another number becomes a sq-
uare, is divided by an optional number and then incre-
ased or decreased by it. The square of half the result
diminished or increased by the greater or smaller (of
the given number) is the number (required).¹
Expressed in the language of algebra, shall have :
= ½ { ½ ( (a - b)/m ± m ) }² ∓ a

1. याभ्यां कृतिरधिको नस्तदन्तरं हृत युतो न मिष्टेन ।
   तद्दल कृतिरधिकोऽधिकयो रविको न यो राशिः ॥ —BrSpSi. XVIII. 74

DOUBLE FIRST DEGREE EQUATIONS 271
or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b
where m is an arbitrary number.
Datta and Singh has given the rationale of this method as
follows :
u² = x ± a; v² = x ± b,
From them, we have u² - v² = ± a ∓ b
Therefore u - v = m
and u + v = (± a ∓ b) / m ,
where m is arbitrary. Hence
u = ½ ( (± a ∓ b) / m + m ) = ± ½ ( (a - b) / m ± m )
Sincc it is obviously immaterial whether u is taken as posi-
tive or negative, we have
u = ½ ( (a - b) / m ± m )
Similarly v = ½ ( (a - b) / m ∓ m )
Therefore x = { ½ ( (a - b) / m ± m ) }² ∓ a,
or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b,
where m is an arbitrary number.
Now we shall take up another particular case, for which
Brahmagupta has given a rule :
The sum of the two numbers tne addition and sub-
traction of which make another number (severally)
a square, is divided by an optional number and then
diminished by that optional number. The square
of half the remainder increased by the subtractive
number is the number (required)¹.
In the algebraic notations, we shall express it as follows :

1. यैरूनो यैश्च युतो रूपैर्वर्गस्तद्द्वयमिष्ट हृतम् ।
   इष्टोनं तद्दल कृतिरूनाऽभ्यधिका भवति राशिः ॥ —BrSpSi. XVIII. 71

274 BRAHMAGUPTA AS AN ALGEBRAIST
y = 1/a ((ad + bc)/m + b)
if b > c and m > (ad + bc)/m. If these conditions be reversed then x
and y will have their values interchanged.
Datta and Singh have given the following rationale of
these solutions :
axy = bx + cy + d,
or a²xy − abx − acy = ad,
or (ax − c) (ay − b) = ad + bc.
Suppose ax − c = m, a rational number;
then ay − b = (ad + bc)/m.
Therefore
x = 1/a (m + c)
y = 1/a ((ad + bc)/m + b)
Or, we may put ay − b = m;
in that case, we shall have ax − c = (ad + bc)/m;
whence x = 1/a ((ad + bc)/m + c),
y = 1/a (m + b).
Brahmagupta’s own rule.
Whilst the rule given above is ascribed to an unknown
author, Brahmagupta’s own rule for the solution of a quadratic
indeterminate equation involving a factum is as follows :
With the exception of an optional unknown, assume
arbitrary values for the rest of the unknowns, the
product of which forms the factum. The sum of the
product of these (assumed values) and the (respective)
coefficients of the unknowns will be absolute quantities.
The continued products of the assumed values and of
the coefficient of the factum will be the coefficient of
the optionally (left out) unknown. Thus the solution

BRAHMAGUPTA'S OWN RULE 275
is effected without forming an equation of the factum
Why then was it done so ?¹
Datta and Singh think that the reference in the latter por-
tion of this rule is to the method of the unknown author :
“Kiṁ kṛtaṁ tadataḥ” ? The principle underlying Brahma-
gupta's method is to reduce, like the Greek Diophantus (c.275
A.D.), the given indeterminate equation to a simple determi-
nate one by assuming arbitrary values for all the unknowns ex-
cept one. So undoubtedly it is inferior to the earlier method.
We now take an illustrative example from Brahmagupta :
On subtracting from the product of signs and degrees of
the Sun, three and four times (respectively) those
quantities, ninety is obtained. Determining the Sun
within a year (one can pass as a proficient) mathema-
tician.
If we presume x to denote the signs and y the degrees of
the Sun, then the equation would be :
xy - 3x - 4y = 90
Pṛthūdaka Svāmī solves it in two ways :
(i) Let us assume the arbitrary number to be 17. then
x = 1/1 ( (90.1 + 3.4) / 17 + 4 ) = 10
y = 1/1 (17 + 3) = 20
(ii) Let us assume arbitrarily y = 20. On substituting
this value of y in the above equation, we get
20x - 3x = 170
whence x = 10.

1. भावितके यद्घातो विनष्टवर्णेन तत्प्रमाणानि ।
   कृतेष्टानि तदाहत वर्णैक्यं भवति रूपाणि ॥
   वर्ण प्रमाणभावित घातो भर्वातेष्ट वर्ण संख्यैवन् ।
   सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥ —BrSpSi. XVIII. 62-63
2. भानो राश्यंशवधात् त्रिचतुर्गुणितान् विशोध्य राश्यंशान् ।
   नवतिं दृष्ट्वा सूर्यं कुर्वन्नावत्सराद् गणकः ॥ —BrSpSi. XVIII. 61.
   — : o : —
   Reference
   H.T. Colebrooke : Algebra with Arithmetic and Mensuration
   from the Sanscrit of Brahmagupta and Bhas-
   cara, London, 1817.
   B. Datta and A.N. Singh : History of Hindu Mathematics Pt. I
   and II, 1962.