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

200 BRAHMAGUPTA AS AN ALGEBRAIST
portion of the equation whilst its other part is practically invisible
or unknown¹.
Unknowns and Symbolism
Āryabhaṭa I (499 A. D.) probably used coloured gulikās or
shots for represtnting different unknowns. Brahmagupta men-
tions varṇa as the symobls for unknown. He has, however, not
indicated how these varṇas or colours were used as symbols for
unknowns. Perhaps we might conculude from this that the
method of using colours as symbols for unknown quantities was
very common and familiar to the algehraists. Datta and Singh
say that the Sanskrit word varṇa denotes colour as well as a letter
of alphabet, and therefore, letters of alphabet came into use for
unknown quantities : kālaka (black), nīlaka (blue), pītaka (yellow),
lohita (red), haritaka (green), śvetaka (white), citraka (variegated),
kapilaka (tawny), piṅgalaka (reddish-brown), dhūmraka (smoke-
coloured), pāṭalaka (pink), śavalaka (spotted), śyāmalaka (blackish)
mecaka (dark blue) etc².
It should be further noted that the first unknown quantity
yāvat-tāvat is not a varṇa or colour. It thus clearly indicates
that the use of colours as symbols came at a later stage, whilst the
word yāvat-tāvat was in currency from much earlier times. Some
authorities think that the term yāvat-tāvat is a corrupted form of
yāvakastāvat (where yāvaka means red). Pṛthūdaka Svāmī has
sometimes used the term yāvaka for an unknown quantity³.
Laws of Signs
Brahmagupta has in his Chaper XVIII devoted a special
section entitled "Dhanarṇa Śunyānāṁ Saṁkalanam" or calcu-
lations dealing with quantities bearing positive and negative signs
and zero,

1. अव्यक्तान्तरं भक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः । —BrSpSi XVIII. 43
   वर्गे चतुर्गुणितानां रूपाणां मध्यवर्गे सहितानाम् ॥ —BrSpSi XVIII. 44
2. यावत्-तावत् कालको नीलकोऽन्यो वर्णः पीतो लोहितश्चैतदाद्याः ।
   अव्यक्तानां कल्पिता मानसंज्ञास्तत्संख्यानं कर्तुंमाचार्यवर्यैः ॥
   यावत्-तावत् कालक नीलक पीताश्च लोहितो हरितः ।
   श्वेतक चित्रक कपिलक पाटलकाः पण्डु धूम्र शवलाश्च ॥
   श्यामलक-मेचक-धवलक-पिशङ्ग-शारङ्ग-बभ्रु-गौराद्याः । —Nārāyaṇa, Bījagaṇita
3. BrSpSi XII, 15. (Com.) ; XII 18.

: Does it look like ऋ? No, it has a horizontal top bar, a round left belly with a notch, connected to a vertical stem on the right: THAT IS 'ध'! Wait, why does it have 'ध'? Could it be ऋणयोर्धनयोर्घातो धनं ऋणयोर्वधो...? Wait, let's re-read the first half: Does it say ऋणयोर्धनयोर्घातो धनं? Wait! Look at the space in: ऋणयोर्धनयोर्घातोIs there a space afterघातो? Yes! Then what is the next letter? ध! Then न! Wait, has it got an anusvara? Look at the top of न: Wait, does it have an anusvara? NO! There is no anusvara on that न! It's just धन! Wait, then मृ? Wait, look at: ऋणयोर्धनयोर्घातो धनमृणयोर्वधोWait, why would it sayधनमृणयोर्वधो धनं भवति`? That would mean product of positive and negative is positive!
Wait, could it be:
"ऋणयोर्धनयोर्घातो धनं, ऋणयोर्वधो धनं भवति"?
Wait! If:
"The product of two negatives is positive; positive multiplied

202 BRAHMAGUPTA AS AN ALGEBRAIST
Brahmagupta lays down the rules regarding evolution and
involution as follows :
The square of a positive or a negative number is positive
.....The (sign of the) root is the same, as was that from
which the square was derived¹.
As regards the latter portion of this rule, Pṛthūdaka Svāmī
has the following comment to make : "The square-root should be
taken either negative or positive, as will be most suitable for
subsequent operations to be carried on."
It will be interesting to observe the following observation
of Mahāvīra (850 A. D.) regarding square-root of a negative
quantity "Since a negative number by its own nature is not a
square, it has no square-root."² So says Śrīpati : "A negative
number by itself is non-square, so its square-root is unreal; so the
rule (for the square-root) should be applied in the case of a posi-
tive number."³
Algebraic Operations
Brahmagupta and other algebraists recognise six operations
as fundamental in algebra : addition, subtraction, multiplication,
division, squaring and the extraction of the square-root.
Regarding addition and subtraction Brahmagupta says :
Of the unknowns, their squares, cubes, fourth powers,
fifth powers, sixth powers, etc., addition and subtraction
are (performed) of the like; of the unlike (they mean
simply their) statement severally.⁴
In place of "of the like", Bhāskara II uses the term "of those
of the same species (jāti) amongst unknowns" :
Addition and subtraction are performed of those of the
same species (jāti) amongst unknowns; of different species
they mean their separate statement.⁵

1. खोद्धृत मृणं धनं वा तच्छेदं खमृणधनविभक्तं वा ।
   ऋणधनयोर्वर्गः स्वं खं खस्य पदं कृतैर्यत् तत् ॥ —BrSpSi, XVIII. 35
2. GSS. I, 52.
3. Śiśe, XIV, 5.
4. अव्यक्तवर्ग घनवर्ग वर्ग पंचगत षड्गतादीनाम् ।
   तुल्यानां संकलित व्यवकलिते पृथगतुल्यानाम् ॥ —BrSpSi.XVIII. 41.
5. योगोऽन्तरं तेषु समान जात्योर्विभिन्न जात्योश्च पृथक् स्थितिश्च ।
   — Bhāskara II, Bījagaṇita.

ALGEBRAIC OPERATIONS 203
This means that the numerical coefficients of x cannot be
added to or subtracted from the numerical coefficients of y
or x² or x³ or xy and so on because these terms belong to different
jāti or they do not belong to the category of the "like".
Again, regarding multiplication, Brahmagupta says :
The product of the two like unknowns is a square; the
product of three or more like unknowns is a power of
that designation. The multiplication of unknowns of
unlike species is the same as the mutual product of
symbols; it is called bhāvita (product or factum).¹
Having given the rules of the operations for addition, sub-
traction and multiplication, Brahmagupta does not think
it necessary to deal with other operations. His section on the
calculations with zero, negative and positive quantities ends here.
How is an Equation Formed?
Pṛthūdaka Svāmī while commenting on a verse in Brāhma-
sphuṭasiddhānta speaks as follows :
In this case, in the problem proposed by the questioner,
yāvat-tāvat is put for the value of the unknown quantity.
Then performing multiplication, division etc. as required
in the problem the two sides shall be carefully made
equal. The equation being formed in this way, then
the rule (for its solution) follows.²
Plan for Writing Equations
When in regards to a given problem, an equation has been
formed, it has to be written down for further operations.
This writing down of an equation is technically known
as nyāsa Perhaps the oldest record of nyāsa is to be found in the
Bakhshālī Manuscript. According to the procedure prescribed in
this work, the two sides of an equation are put down one after the
other in the same line without any sign of equality being inter-
posed. Thus the equations :
√(x+5)=s, √(x-7)=t
appear as

1. सदृशद्विवधो वर्गस्त्र्यादिवधस्तद् गतोऽन्यजातिवधः ।
   अन्योन्यवर्णघातो भावितकः पूर्ववच्छेदम् ॥ —BrSpSi.XVIII. 42.
2. BrSpSi. XVIII. 43 (com)

204 BRAHMAGUPTA AS AN ALGEBRAIST
| 0 5 yu mū 0 | sa 0 7 + mū 0 |
| 1 1 1 | 1 1 1 |
Here yu (यु) stands for yuta (युत), meaning added, subtraction
is+sign, derived from Kṣaya or (क्षय) meaning diminished, gu
(गु) for guṇa or guṇita, meaning multiplied; bhā (भा) for division
from bhājita and mū (मू) for square-root, from mūla meaning
root; zero (०) was used to mark a vacant place.
Again, the following equation
x+2x+3×3x+12×4x=300
is represented as ;
| 0 | 2 1 | 3 3 | 12 4 | dṛśya 300
| 1 | 1 1 | 1 1 | 1 1 |
There is no sign for unknown in the Bakhshālī Manuscript.
Later on this plan of writing equations as adopted in Bakha-
śālī Manuscript was abandoned in India; a new one was adopted
in which the two sides are written one below the other without
any sign of equality. It must be stated that in this new plan the
term of similar denominations are usually written one below the
other and even the terms of absent denominations on either side
are clearly indicated by putting zeros as their coefficients. We
find a reference to this new plan in the algebra of Brahmagupta.
From which the square of the unknown and the unkno-
wn are cleared, the known quantities are cleared (from
the side) below that¹.
Here in this verse, the words “adhastāt” clearly indicate
that one side of the equation is written below the other., As an
illustration, Pṛthūdaka Svāmī represented the equation² :—
10x-8=x²+1
as :
ya va 0 ya 10 ru 8̇ (x².0+x.10-8)
ya va 1 ya 0 rū 1 (x².1+x. 0+1)
which means, x² was written as yāvat-varga (yā va) and
x was written as yāvat or yā. The minus sign was represented
by a dot at the top of the number.(-8 was written as 8̇). We
shall take another illustration from Pṛthūdaka Svāmī.
He would write the equation
197x-1644 y-z=6302
as

1. BrSpSi. XVII. 43; compare also Bhāskara II, Bījagaṇita.
2. Br SpSi. XVIII. 49 (com.)

PLAN FOR WRITING EQUATIONS 205
yā 197 kā 1644̇ nī 1̇ rū 0
yā 0 kā 0 nī 0 rū 6302
Here the first unknown x is represented by yā(vat), the
second unknown y by kā(laka) and the third unknown z by nī-
(laka) and the term without unknown, a mere number is written
by rū(paka). The two sides, one written below the other if
written in the present form, would appear as :
197x—1644y—z+0=0x+0y+0z+6302.
The Bījagaṇita of Bhāskara II also follows the same proce-
dure. One instance from it would be quoted here to illustrate
the method of expressing equations.
8x³+4x²+10y²x=4x³+12y²x
or 8x³+4x²+10y²x=4x³+0x²+12y²x
is written as follows on Bhāskara's or Brahmagupta's plan :
x³ is ghana of yāvat (abbreviated as yā gha)
x² is varga of yāvat (abbreviated as yā va)
y² is varga of kālaka (abbreviated as kā va)
the coefficients 10 and 12 are bhāvita (abbreviated as
bhā).
The equation is :
yā gha 8 yā va 4 kā va yā. bhā 10
yā gha 4 yā va 0 kā va yā. bhā 12
Datta and Singh state that the use of the old plan of writ-
ing equations is sometimes met with in later works also. For
instance, in the MS. of Pṛthūdaka Svāmī's commentary¹ on the
Brāhmasphuṭasiddhānta, an incomplete copy of which is preserv-
ed in the library of the Asiatic Society of Bengal (No. I B6), we
find a statement of equations thus : “first side yāvargaḥ 1 yāvakaḥ
200 rū 0; second side yāvargaḥ 0 yāvakaḥ 0 rū 1500.
Śodhana or Clearance of an Equation
After nyāsa or statement of an equation, the operation to
be performed is known as śodhana (clearance) or saṁśodhana
(equi-clearance or complete clearance). The nature of
this clearance varies according to the kind of equation
In the case of an equation in one unknown only, whether linear,

1. BrSpSi. XII. 15 (com.)

206 BRAHMAGUPTA AS AN ALGEBRAIST
quadratic or of higher powers, one side of it is cleared of the
unknowns of all denominations and the other side of it of the
absolute terms, so that the equation is ultimately reduced to one
of the form
ax² + bx = c,
where a, b, c may be positive or negative; some of them may
even be zero. Thus Brahmagupta observes :
From which the square of the unknown and the un-
known are cleared, the known quantities (rūpāṇi) are
cleared (from the side) below that.¹
On this Pṛthūdaka Svāmī comments as follows :
This rule has been introduced for that case in which
the two sides of the equation having been formed in
accordance with the statement of the problem, there are
present the square and other powers of the unknown
together with the (simple) unknown. The absolute
terms should be cleared off from the side opposite to
that from which are cleared the square (and other
powers) of the unknown and the (simple) unknown.
When perfect clearance (saṁśodhana) has been thus
made...²
Śrīdhara and Bhāskara II have also given the rules of
clearance almost on the same lines. Thus the equation
yā va 0 yā 10 rū 8̇
yā va 1 yā 0 rū 1
after perfect clearance having been made will be (according to
Pṛthūdaka Svāmī)
yā va 1 yā 10 rū 9̇
i.e. the equation 10x - 8 = x² + 1
after clearance would become
x² - 10x = -9.
Classification of Equations
Usually equations are classified as :
simple equation : yāvat-tāvat
quadratic : varga

1. वर्गाव्यक्ताः शोध्या यस्माद् रूपाणि तदधस्तात् ॥ — BrSpSi. XVIII. 43.
2. BrSpSi. XVIII. 43 (com.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1. Mahāsiddhānta. Bhāskara II, XV. 50

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

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