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

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

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