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

190 BRAHMAGUPTA AS AN ALGEBRAIST
A.D.). There can be no doubt regarding the age of these two
authors. Bhāskara II completed his great work on the Siddhānta-
śiromaṇi in 1072 Śaka, and Karaṇa-kutūhala a practical astro-
nomical treatise in 1105 Śaka; these dates are based on the
passages given by Bhāskara himself in his works. The Bīja-
gaṇita and the Līlāvatī form parts of the great treatise, the
Siddhānta-śiromaṇi. The genuineness of the text is established,
as Colebrooke says, with no less certainty by numerous commen-
tators in Sanskrit, besides a Persian version of it. Those com-
mentaries comprise a perpetual gloss, in which every passage of
the original is noticed and interpreted : and every word of it is
repeated and explained. From comparison and collation of
various texts, it appears then that the work of Bhāskara, exhibit-
ing the same uniform text which the modern transcripts of it
do, was in the hands of both Muhammedans and Hindus,
between two or three centuries ago : and numerous copies of it
having been diffused throughout India, at an earlier period, as of
a performance held in high estimation, It was the subject of
study and habitual reference in countries and places so remote
from each other as the north and west of India and the Southern
Peninsula.
This though not marking any extraordinary antiquity, nor
approaching to that of the author himself, was a material point
to be determined : as there will be in the sequel, so says Cole-
brooke, occasion to show, that modes of analysis, and in parti-
cular, general methods for the solution of indeterminate prob-
lems both of the first and second degrees, are taught in the
Bīja-gaṇita, and those for the first degrees repeated
in the Līlāvatī, which were unknown to the mathematicians of
the West, until invented anew in the last two centuries by
algebraists of France and England.¹ Bhāskara who himself
flourished more than six hundred and fifty years ago, was in
this respect a compiler and took those methods from Indian
authors as much more ancient than himself.
Regarding the age of the precursors of Bhāskara II, Cole-
brooke says : The age of his precursors cannot be determined
with equal precision. He then proceeds to examine the evidence
as follows :

1. Colebrooke, H. T., Miscellaneous Essays, p. 421.

ALGEBRA GOES TO EUROPE FROM INDIA 191
Towards the close of his treatise on Algebra, Bhāskara
II informs us, that it is compiled and abridged from
the more diffuse works on the same subject, bearing
the names Brāhme (meaning no doubt Brahmagupta),
Śrīdhara and Padmanābha; and in the body of his
treatise, he has cited a passage of Śrīdhara's algebra
and another of Padmanābha. He repeatedly adverts
to preceding writers and refers to them in general
terms, where his commentators understand him to
allude to Āryabhaṭa, to Brahmagupta to the latter's
scholiast Caturveda Pṛthūdaka Svāmī and to the other
writers above mentioned.
Most, if not all, of the treatises, to which he thus
alludes, must have been extant, and in the hands of
his commentators, when they wrote; as appears from
their quotations of them; more especially those of
Brahmagupta and Āryabhaṭa, who are cited, and
particularly the first mentioned, in several instances.
A long and diligent research in various parts of India,
has, however, failed of recovering any part of the
Padmanābha Bīja (or the algebra of Padmanābha) and
of the algebraic and other works of Āryabhaṭa.
But the translator has been more fortunate in regard
to the works of Śrīdhara and Brahmagupta, having in
his collection Śrīdhara's compendium of arithmetic,
and a copy incomplete however, of the text and
scholia of Brahmagupta's Brahmasiddhānta comprising
among other no less interesting matter, a chapter
treating of arithmetic and mensuration; and another,
the subject of which is algebra : both of them fortu-
nately complete. The commentary is a perpetual one;
successively quoting in length each verse of the text;
proceeding to the interpretation of it, word by word;
and subjoining elucidations and remarks; and its colo-
phon, at the close of each chapter, gives the title
of the work and the name of the author. Now the
name which is there given, Caturveda Pṛthudaka
Svāmī, is that of a celebrated scholiast of Brahmagupta,
frequently cited as such by the commentaries of

192 BRAHMAGUPTA AS AN ALGEBRAIST .
Bhāskara and by other astronomical writers; and the
title of the work, Brāhmasiddhānta or sometimes Brāh-
masphuṭasiddhānta, corresponds, in the shorter form,
to the known title of Brahmagupta's treatise in the
usual references to it by Bhāskara's commentators, and
answers, in the longer form, to the designation of it, as
indicated in an introductory couplet which is quoted
from Brahmagupta by Lakṣmīdāsa, a scholiast of Bhās-
kara II. Remarking this coincidence, the translator
proceeded to collate, with the text and commentary,
numerous quotations from both, which he found in
Bhāskara's writings or in those of his expositors. The
result confirmed the indication and established the
identity of both text and scholia as Brahmagupta's
treatise, and the gloss of Pṛthūdaka. The authenticity
of the Brāhmasiddhānta is further confirmed by numer-
ous quotations in the commentary of Bhaṭṭotpala on
the Saṁhitā of Varāhamihira : as the quotations from
the Brāhmasiddhānta, in that commentary, (which is
the work of an author who flourished eight hundred
and fifty years ago) are verified in the copy under
consideration. A few instances of both will suffice,
and cannot fail to produce conviction.
It is confidently concluded, that the chapters on arith-
metic and algebra, fortunately entire in a copy in many
parts imperfect, of Brahmagupta's celebrated work as
here described, are genuine and authentic. It remains
to investigate the age of the author.
Mr. Davis, who first opined to the public a correct
view of the astronomical computations of the Hindus,
is of opinion, that Brahmagupta lived in the seventh
century of the Christian era. Dr. William Hunter, who
resided for some time with a British Embassy at Ujjay-
inī, and made diligent researches into the remains of
Indian science at that ancient seat of Hindu astrono-
mical knowledge, was there furnished, by the learned
astronomers whom he consulted, with the ages of the
principal ancient authorities. They assigned to Brah-
magupta the date of 550 Saka; which answers to A.D.

ALGEBRA GOES TO EUROPE FROM INDIA 193
628. The grounds on which they proceeded are unfor-
tunately not specified : but as they gave Bhāskara's
age correctly, as well as several other dates right,
which admit of being verified; it is presumed that they
had grounds, though unexplained, for the information
which they communicated..
Mr. Bentley, who is little disposed to favour the anti-
quity of an Indian astronomer, has given his reasons
for considering the astronomical system which Brah-
magupta teaches, to be between twelve and thirteen
hundred years old (1263 years in A.D. 1799). Now as
the system taught by this author is professedly one
corrected and adapted by him to conform with the
observed positions of the celestial objects when he
wrote, the age, when their positions would be conform-
able with the results of computations made as by him
directed, is precisely the age of the author himself :
and so far as Mr. Bentley's calculations may be consi-
dered to approximate the truth, the date of Brahma-
gupta's performance is determined with like approach .
to exactness, within a certain latitude however of
uncertainty for allowance to be made on account
of the inaccuracy of Hindu observations.
The translator has assigned on former occasions the
grounds upon which he sees reason to place the author's
age, soon after the period when the vernal equinox
coincided with the beginning of the lunar mansion and
zodiacal asterism Aśvini, where the Hindu ecliptic
now commences. He is supported in it by the senti-
ments of Bhāskara and other Indian astronomers, who
infer from Brahmagupta's doctrine concerning the
solistitial points, of which he does not admit a periodi-
cal motion, that he lived when the equinoxes did not,
sensibly to him, deviate from the beginning of Aśvini
and middle of citrā on the Hindu sphere. On these
grounds it is maintained, that Brahmagupta is rightly
placed in the sixth or beginning of the seventh century
of the Christian era, as the subjoined calculations will
more particularly show. The age when Brahmagupta

194 BRAHMAGUPTA AS AN ALGEBRAIST
flourished, seems then, from the concurrence of all these
arguments, to be satisfactorily settled as antecedent
to the earliest dawn of the culture of sciences among
the Arabs; and consequently establishes the fact, that
the Hindus were in possession of algebra before it was
known to the Arabians.
Brahmagupta's treatise, however, is not the earliest
work known to have been written on the same subject
by an Indian author. The most eminent scholiast of
Bhāskara II (Gaṇeśa) quotes a passage of Āryabhaṭa
specifying algebra under the designation of Bīja, and
making separate mention of Kuṭṭaka, which more par-
ticularly intends a problem subservient to the general
method of resolution of indeterminate problems of the
first degree : he is understood by another of Bhāskara's
commentators to be at the head of the elder writers, to
whom the text then under consideration adverts, as
having designated by the name of Madhyamāharaṇa
the resolution of affected quadratic equations by means
of the completion of the square. It is to be presumed,
therefore, that the treatise of Āryabhaṭa then extant,
did extend to quadratic equations in the determinate
analysis, and to indeterminate problems of the first
degree; if not to those of the second likewise, as most
probably it did.
This ancient astronomer and algebraist, so says Cole-
brooke, was anterior to both Varāhamihira and
Brahmagupta, being repeatedly named by the latter;
and the determination of the age when he flourished
is particularly interesting as his astronomical system,
though on some points agreeing, essentially dis-
agreed on others, with that which the Hindu astrono-
mers still maintain.
He, as Colebrooke says, is considered by the commen-
tators of the Sūryasiddhānta and Śiromaṇi, as the
earliest of uninspired and mere human writers on the
science of astronomy, as having introduced requisite
corrections into the system of Parāśara, from whom he
took the numbers for the planetary mean motions; as

ALGEBRA GOES TO EUROPE FROM INDIA 195
having been followed in the tract of emendation,
after a sufficient interval to make further correction
requisite, by Durgāsinha and Mihira; who were again
succeeded after a further interval by Brahmagupta, son
of Jiṣṇu.
In short, says Colebrooke, Āryabhaṭa was founder of
one of the sets of Indian astronomers, as Puliśa, an
author likewise anterior to both Varāhamihira and
Brahmagupta, was of another : which were distingui-
shed by names derived from the discriminative tenets
respecting the commencement of planetary motions at
sunrise according to the first, but at midnight accord-
ing to the latter, on the meridian of Laṅkā, at the
beginning of the great astronomical cycle. A third
sect began the astronomical day, as well as the great
period, at noon.
Āryabhaṭa’s name accompanied the intimation which
the Arab astronomers (under the Abbasside Khalifs, as
it would appear,) received, that three distinct astro-

• nomical systems were current among the Hindus of
  those days : and it is but slightly corrupted, certainly
  not at all disguised, in the Arabic representation of
  it Arjabahar, or rather Arjabhar, (corrupted form of
  Āryabhaṭa). The two other systems were, first,
  Brahmagupta’s Siddhānta which was the one they
  became best acquainted with, and to which they apply
  the denomination of the sind-hind; and second, that of
  Arca, the Sun, which they write Arcand a corruption
  still prevalent in the vulgar Hindi.
  Āryabhaṭa appears to have had more correct notions
  of the true explanation of celestial phenomena than
  Brahmagupta himself, so says Colebrooke; who in a
  few instances, correcting errors of his predecessor, but
  oftener deviating from that predecessor’s juster views,
  has been followed by the herd of modern Hindu astro-
  nomers, in a system not improved, but deteriorated,
  since the time of the more ancient author.
  Considering the proficiency of Āryabhaṭa in astronomi-
  cal science, and adverting to the fact of his having

196 BRAHMAGUPTA AS AN ALGEBRAIST
written algebra, as well as to the circumstance of his
being named by numerous writers as the founder of a
sect, or author of a system in astronomy, and being
quoted at the head of algebraists, when the commen-
tators of extant treatises have occasion to mention
early and original writers on this branch of science, it
is not necessary to seek further for a mathematician
qualified to have been the great improver of the ana-
lytic art, and likely to have been the person by whom
it was carried to the pitch to which it is found to have
attained among the Hindus, and at which it is observ-
ed to be nearly stationary through the long lapse of
ages which have since passed : the later additions being
few and unessential in the writings of Brahmagupta,
of Bhāskara and of Jñānarāja, though they lived at
intervals of centuries from each other.
Āryabhaṭa, Colebrooke rightly says, then being the
earliest author known to have treated of Algebra among
the Hindus, and being likely to be, if not the inventor,
the improver of that analysis, by whom too it was
pushed nearly to the whole degree of excellence which
it is found to have attained among them; it becomes
in an especial manner interesting to investigate any
discoverable trace in the absence of better and more
direct evidence, which may tend to fix the date of his
labours; or to indicate the time which elapsed between
him and Brahmagupta, whose age is more accurately
determined.
Taking Āryabhaṭa, for reasons given, to have preceded
Brahmagupta and Varāhamihira by several centuries;
and Brahmagupta to have flourished more than twelve
hundred years ago, and Varāhamihira, concerning
whose works and age, Colebrooke has given a few
notes, and has placed him at the beginning of the sixth
century after Christ, it appears probable that this
earliest of known Hindu algebraists wrote as far back
as the fifth century of the Christian era; and perhaps in
an earlier age. Hence it is concluded that he is nearly
as ancient as the Grecian algebraist Diophantus, sup-

ALGEBRA GOES TO EUROPE FROM INDIA 197
posed on the authority of Abulfaraj, to have flourished
in the time of the Emperor Julian or about A. D.
360.
Colebrooke further says : Admitting the Hindu and
Alexandrian authors to be nearly equally ancient, it
must be conceded in favour of the Indian algebraist,
that he was more advanced in the science; since he
appears to have been in possession of the resolution of
equations involving several unknowns, which it is not
clear, nor fairly presumable, that Diophantus, knew;
and a general method of indeterminate problems of at
least the first degree, to a knowledge of which the
Grecian algebraist had certainly not attained ; though
he displays infinite sagacity and ingenuity in particu-
lar solutions ; and though a certain routine is indiscer-
nible in them.
Colebrooke appears to be of the view that Greeks.
were the first to discover the solution of equations
involving one unknown; and this knowledge was passed
to ancient Indians by their Greek instructors in impro-
ved astronomy. But "by the ingenuity of the Hindu-
scholars, the hint was rendered fruitful and the algeb-
raic method was soon ripened from that slender beginn-
ing to the advanced state of a well arranged science,
as it was taught by Āryabhaṭa, and as it is found in
treatises compiled by Brahmagupta and Bhāskara."
We do not agree with this analysis in entirety. Indian
algebra is entirely of Indian roots. It had its beginning
in the times of Saṃhitās and Brāhmaṇas. Some of the
equations and problems were solved by geometric
methods. It must have had its origin in the Śulba
period if not before. Āryabhaṭa undoubtedly was the
discoverer of many algebraic solutions of equations of
the first and higher order with one and more unkno-
wns. It is rather too much to trace the influence of
Diophantus on Indian algebra which developed in this
country independently. Brahmagupta is one of the
most brilliant algebraists we ever had in the entire
history of mathematics.

198 BRAHMGUPTA AS AN ALGEBRAIST
Technical Terms
Coefficient—
In the ancient Indian algebra, there is no systematic term
for the coefficient. Usually, the power of the unknown is men-
tioned when the reference is to the coefficient of that power. At
one place, for example, we find Pṛthūdaka Svāmī (the commenta-
tor of Brahmagupta's Brāhmasphuṭasiddhānta) writing “the num-
ber (aṅka) which is the coefficient of the square of the unknown is
called the ‘square’ and the number which forms the coefficient of
the (simple) unknown is called the ‘unknown quantity’ (avyakta-
māna).”¹ However, at many places, we find the use of a technical
term also. Brahmagupta once calls the coefficient saṁkhyā²
(number) and on several other occasions guṇaka³ or guṇakāra⁴
(multiplier). Pṛtthūdaka Svāmī (860 A.D.) calls it aṅka (number)
or prakṛti (multiplier). These terms may also be seen in the
works of Śrīpati⁶ (1039) and Bhāskara II⁷ (1150 A.D.). The
former also used the word rūpa for the same purpose.⁸
Unknown Quantity
The unknown quantity has been termed as yāvat-tāvat
(meaning so-much-as or as-many-as) in literature as early as 300
B. C. (vide the Sthānāṅga-sūtra⁹). In the Bakhasālī Manuscript,
it has been termed as yadṛcchā, vāñchā or kāmikā (or any desired
quantity)¹⁰. Āryabhaṭa I in one of his verses calls the unknown
as gulikā¹¹ (literally meaning a shot) From the early seventh
century A.D., the word avyakta was used for unknown quantities.
Brahmagupta uses this term in his Brāhmasphuṭasiddhānta¹²

1. BrSpSi. XVIII. 44 (Com.)
2. वर्ण्यप्रमाण भावितघातो भवत्तीष्ट वर्णं संख्येवम् ! —BrSpSi. XVIII 63
3. मूलं द्विषेष्ट वर्गाद् गुणक गुणादिष्ट युत विहीनाच्च । —BrSpSi. XVIII 64
   वर्गच्छिन्ने गुणके प्रथमं तन्मूल भाजितं भवति । —BrSpSi. XVIII 70
4. प्रथमोऽन्यमूलमन्यो गुणकार पदोद्धृतः प्रथमः । —BrSpSi. XVIII 69
5. BrSpSi XVIII. 44 (Com.)
6. SiSe XIV. 33-5.
7. Bījagaṇita
8. SiSe XIV. 33-5.
9. Sūtra 747.
10. BMs. Folio 22, verso; 23, recto and verso.
11. गुलिकान्तरेण विभजेद् द्वयोः ।
12. अव्यक्तवर्गं धनवर्गं वर्गपञ्चगत षड्गतादीनाम् ।

TECHNICAL TERMS 199
Power
Since long, the word varga has been used for the second
power; the word also stands for square (Uttarādhyayana Sūtra¹,
B. C. c. 300). The third power is similarly known as ghana, the
fourth power as varga-varga (square-square), the sixth power as
ghana-varga (cube-square) and the twelfth power as ghana-varga-
varga (cube-square-square). In later days, the fifth power was
called vargaghana ghāta (here the word ghāta means product; the
term means product of cube and square). The former system was
multiplicative, rather than additive; wheras the latter was on the
additive system. The seventh power on the additive system was
known as varga-varga-ghana-ghāta (product of square-square and
cube). Brahmagupta, however, uses a more scientific system for
expressing the powers more than four. He calls the fifth power
as Pañca gata (literally meaning, raised to the fifth), the sixth
power as ṣaḍ-gat (raised to the sixth) and so on, thus adding the
suffix gata to the name of the number indicating that power.²
Bhāskara II has followed the system of Brahmagupta almost consi-
stently for powers one and upwards.
Equation
Perhaps Brahmagupta has for the first time used the term
samakaraṇa or samīkaraṇa (literally meaning making equal) or
simply sama (equal or equation)³. Pṛthūdaka Svāmī (860)
employs the term sāmya (equality or equation) for equation⁴.
The equation is said to possess two Pakṣas⁵ (sides) Itara-Pakṣa
and apara-pakṣa.
Absolute Term
Brahmagupta uses the term rūpa (literally meaning appear-
ance) for an absolute term. It represents the visible or known

1. Chapter XXX, 10, 11.
2. अव्यक्तवर्गे घनवर्ग-वर्ग-पंचगत-षड्गतादीनाम् ।
   सदृश द्विवधो वर्गस्त्यादि वधस्तद्गतोऽन्य जातिवधः ॥
3. वर्णं प्रमाण भावित-घातो भवतीष्टवर्ण संख्यैवम् ।
   सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥
   अव्यक्तान्तर भक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः ।
4. Siśe. XIV, 19.
5. Bījagaṇita.

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