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

148 EPICYCLIC THEORY OF ANCIENT INDIANS
Let E represent the centre of the Earth(Fig.15).APM the Sun's
circular orbit or concentric; let A and P be the apogee and the
perigee respectively. From EA, cut off EC equal to the radius of
the Sun's epicycle. With centre C and radius equal to EA describe
the eccentric A'P'S cutting AP and AP produced at P' and A'.
Here A' and P' are the real apogee and perigee of the Sun's orbit.
Let PM and P'S be any two equal arcs measured from P and P'.
The idea is that the mean planet M and the apparent Sun S
move simultaneously from P and P' in the counterclockwise
direction along the concentric and the eccentric circles. They
move with the same angular motion and arrive simultaneously
at M and S.
Here EM and CS are parallel and equal, hence MS is
also equal and parallel to EC. Let SH be drawn perpendicular
to EM.
The angle PEM is the mean anomaly and the angle P'ES the
true anomaly; the angle SEM is the equation of the centre, is readily
seen to be plus (+) from P' to A' and minus (—) from A' to P'.
Thus as regards the character of the equation, the eccentric circle
is quite right. We now turn to exmine how far it is true as to
the amount.
Let the angle SEM denoted by E and the angle ∠PEM
=∠P'CS=θ; EP=CP'=a; EC=MS=p, then
tan E = SH / HE = p sin θ / (a - p cos θ)
∴ E = (p / a) sin θ - (p² / 2a²) sin 2θ + (p³ / 3a³) sin 3 θ.........
Now the true value of E in elliptic motion is given by
E = ( 2e - e³/4 ) sin θ + (5/4) e² sin 2θ + (13e³ / 12) sin 3θ*;...
It we now put p / a = 2e - e³/4, as a first approximation p / a = 2e.
Hence p² / 2a² = 2e², which is greater than 5/4 e² by 3/4 e². In the
case of the Sun if the value of p be correctly taken the error in the
coefficient of the second term becomes +3'; similarly in the case of
the Moon, the corresponding error becomes +8'.
*Godfray's Astronomy, p. 149.

APPARENT MOTION OF THE SUN AND MOON 149
Again if p/a=2e, what is the centre of the eccentric circle is
the empty focus of the ellipse or that the ancient astronomers
practically took the planets to be moving with uniform angular
motion round the empty focus. This was not a bad approxi-
mation.
Also ES=r=EH approximately,
∴ r=a ( 1 - p/a cos θ )
but in the elliptic motion
r=a (1-e cos θ).*
Hence the error is not very considerable here also.
This is the way in which the ancient astronomers, both
Greek and Hindu, sought to explain the inequalities in the motion
of the Sun and the Moon. In the case of the Moon, these astro-
nomers took the coefficient 2e - e³/4 = 300' nearly; the modern value
is 377' nearly. The reason for this has been pointed out to be
that the Moon was observed correctly only at times of eclipses.
At the eclipses of syzygies, the evection term of the Moon's equa-
tion diminishes (numerically) the principal ecliptic term by
about 76'.
We have thus far explained the idea of planetary motion of
the ancients under the eccentric circle costruction. The same,
however, is explained under the epicyclic construction.
Let AMP be the circular orbit of the Sun, having E the centre
of the earth for the centre. (Fig. 16)
Let the diameter AEP be the apse
line. A the apogee and P the perigee.
Let M be the mean position of the
Sun in the orbit. With M as the
centre, describe the epicycle UNS.
Let EM cut the epicycle at N and U.
Now the construction for finding S
the apparent Sun is thus given:—
Fig. 16
Make ∠UMS= ∠MEA, the arc US is measured clockwise
whereas the arc A to M is measured counterclockwise.
From this construction MS is parallel to EA. If EC be
measured equal to MS, the radius of the epicycle, along EA to-

• Godfray's Astronomy, p. 149.

150 EPICYCLIC THEORY OF ANCIENT INDIANS
wards the apogee, then CS is a constant length and C is a fixed
point. Hence the locus of S is an equal circle with the centre at
C. Thus both the eccentric, and the epicycle and the concentric
combined, led to the same position.
It was thus usual to explain the planetary motion under
both the assumed constructions; and both gave the position for a
planet. The eccentric circle construction appears to be the earlier
in the history of astronomy and the latter was later. If the
former construction can be traced to Apollonius of Perga who did
so much to develop the "conic sections" as science, the reason why
he preferred the eccentric circle to the ellipse, appears to be that
either that this planetary construction was always deep-rooted in
the minds of men or that he was carried by the idea that "the
circle was the most perfect curve." We are inclined to the view
that the eccentric circle idea was transmitted from Babylonia to
Greece. We now pass on to consider the Indian construction for
the possition of superior and inferior planets.
Superior Planets
With regard to the five planets, Mercury, Venus, Mars,
Jupiter and Saturn, the Indian astronomers give only one constru-
ction for finding the apparent geocentric position. Each of
these "star planets" is conceived as having twofold planetary
inequalities : (i) the inequality of apsis, (ii) the inequality of the
śīghra. With regard to the superior planets, the śīghra apogee or
the śīghrocca coincided with the mean position of the Sun. As
Varāhamihira observed, of the other planets beginning with Mars,
the Sun is the so-called śīghra. (PSi. XVII. 1)
Let AMSP be the concentric of which the centre E is the
same as that of the earth(Fig.17);
A'M₁P' the eccentric circle of
apsis of a superior planet, of
which the centre is C;A,M,S,P be
respectively the apogee, the Mars
.planet, the direction of the
śīghra and the perigee of the
concentric; A',M₁,P' be the
apogee, the planet as corrected
by the equation of apsis, P' the
perigee in the eccentric. The arc
Fig. 17

INFERIOR PLANETS 151
AM=arc A'M₁; MM₁ is parallel and equal to EC. As used be-
fore, both the concentric and the eccentric are of the same radius.
Here the mean planet M in the concentric is taken to be,
deflected to M₁ due to the true motion in the eccentric circle.
Join EM₁ cutting the concentric at M₂. Now let ES be joined
and let S' be taken along ES, such that
ES' / ES = (Śīghra periphery of the planet in degrees) / 360
= (Sun's mean distance from the Earth) / (Planets mean distance from the Sun or the Earth);
ES' thus determined is called the radius of the śīghra epicycle of
the superior planet.
With S' as the centre and the radius equal to ES or EA
describe another circle which is called the śīghra eccentric cutting
ES produced at S''. Now measure the arc S''M₃ in the concen-
tric=SM₂ in the concentric. The apparent superior planet is
seen in the direction EM₃ from the Earth. This is the construc-
tion used in Hindu astronomy calculating the geocentric longitude
of any star planet.
It is evident in the case of a superior planet that the eccen-
tric having S' for the centre and whose radius=EA=R the
standard radius for any circular orbit, is the mean orbit of
the planet and S' the mean position of the Sun. In other words,
in the case of a superior planet, the śīghra eccentric represents
the mean orbit round the Sun. If the parallelogram CES'C' be
constructed, then an equal circle described with C' as the centre
is the apparent eccentric orbit of the superior planet.
In the actual method of calculating the geocentric longitude
of a “star planet” there are four operations given, the first two of
which have the effect of changing the arc MA or rather the point
A. The last two operations relate to the two displacements MM₁
and M₂M₃. We have here followed solely the construction of
the eccentric circles; the same geocentric position of a superior
planet could be equally well obtained by the epicyclic cons-
truction. In describing the constructions for finding the position,
of an inferior planet we shall follow the epicyclic construction
only
Inferior Planets
Let E be the centre of the Earth (Fig. 18), AMS the orbit of

152 EPICYCLIC THEORY OF ANCIENT INDIANS
a mean inferior planet or the mean Sun. EA the direction of the
apogee of apsis and ES that of the
śīghra. The inequality of the apsis takes
the mean geocentric planet from M to
M₁, such that MM₁ is parallel to EA.
Let EM₁ be joined cutting the con-
centric at M₂ ; M₂ is taken as the
centre of the śīghra epicycle or the
real circular orbit in which the appa-
rent planet moves.
Fig. 18
With M₂ as the centre and the radius of the inferior pla-
nents' śīghra epicycle as radius, describe the circle NVU which
is here the śīghra epicycle or the real circular orbit. In it draw
the radius M₂V parallel to ES; then EV is the geocentric position
of the inferior planet.
Here the first displacement MM₁ is due to the inequality
of apsis and is for finding the position of M₂ the centre of the
real circular orbit. The idea was that the apparent planet moved
in a circular orbit of which the centre was very near the mean
position of the Sun, the first operation in this construction was
calculated to determine the centre of this so-called circular orbit
of an inferior planet.
The śīghra of an inferior planet moves round the Earth at
the same mean rate in which the inferior planet moves round the
Sun; hence the line ES in this figure is always parallel to the line
joining the Sun to the mean heliocentric inferior planet, and in
our construction, it is parallel to M₂V.
This in brief is an outline of the Indian idea of planetary
motion as taught by Āryabhaṭa I, Brahmagupta and Bhāskara II.
— : o : —
Reference
P.C. Sengupta : The Khaṇḍakhādyaka, 1934.

CHAPTER VIII
Brahmagupta
and Arithmetic
Scope of Gaṇita
The word Gaṇita means the science of calculation. The
term occurs in the Vedāṅga Jyautiṣa (c. 1200 B. C.) :
Just as the crest is to the peacocks, and just as the head-
gem is to the snakes, so the Gaṇita among the Vedāṅga
Śāstras stands at the head.¹ (VJ. 4)
In the ancient Buddhistic literature, we find mention of
three classes of gaṇita : (i) mudrā (finger arithmetic). (ii)
gaṇana (mental arithmetic), and (iii) saṅkhyāna (higher arith-
metic in general). In the Brāhmasphuṭasiddhānta, Brahmagupta
uses the word gaṇita in the sense of entire calculations. His
gaṇitādhyāya (Chapter XII) includes ;
(i) Miśraka (mixtures). (ii) Śreḍhī (series). (iii) Kṣetra
plane figures), (iv) Vṛtta-kṣetra (circles), (v) Khāta (ex-
cavations), (vi) Citi (piles of bricks), (vii) Krākacika
(sawn pieces of timber), (viii) Rāśi (heaps or mounds of
grain, and (ix) Chāya (shadow).
Brahmagupta also uses the term Dhūlīkarma (literally meaning
“ʇsnpwork”) for higher mathematics :
The one learned man who knows the dhūlīkarma or the
science of mathematics as propounded by Brahmagupta
would far excell them in learning who are taught the
calculations according to Āryabhaṭa, Viṣṇucandra and
others.²
In these ten chapters of the Brahmasiddhānta has been
given the dhūlīkarma or the science of entire calculations

1. यथा शिखा मयूराणां नागानां मणयो यथा ।
   तद् वद् वेदांग-शास्त्राणां गणितं मूर्धनि स्थितम् ॥ VJ. 4.
2. नावायों ज्ञातेरपि तन्त्रैरार्यभटविष्णुचन्द्राद्यैः ।
   यो ब्रह्म धूलिकर्मविदाचार्यत्वं भवति तस्य ॥ —BrSpSi. X. 62.

154 BRAHMAGUPTA AND ARITHMETIC
which is faultless.¹
This science of dhūlīkarma has not been imparted by great
teachers for blasphemy. One who would be using it for this
purpose would lose all good name.²
Brahmagupta uses the term gaṇita only for those calculations
which are of arithmetical in nature. The science of algebra, the
foundations of which was laid by Āryabhaṭa I, was named as
kuṭṭaka or kuṭṭākāra by Āryabhaṭa, and in the Brāhmasphuṭa-
siddhānta also it is separately dealt with under Kuṭṭādhyāya or
kuṭṭakādhāya (Chapter XVIII). Later on the term bījagaṇita
was specifically given to the science of algebra.
The Kuṭṭādhyāya of the Brāhmasphuṭasiddhānta deals with
the (i) concept of kuṭṭaka (pulveriser), addition of positive and
negative as well as zero quantities, equations in one unknown
(eka-varṇa samīkaraṇa), equations in several unknowns (aneka-
varṇa samīkaraṇa), equations involving products of unknowns
(bhāvita) and quadratic equations (varga-prakṛtiḥ) (Chapter XVIII
of the Brāhmasphuṭasiddhānta).
Āryabhaṭa, Bhāskara and
Brahmagupta use Place Value Notations.
In Europe the first definite traces of the place-value nume-
rals are found in the tenth and eleventh centuries, but the
numerals came into general use in mathematical text books only
in the seventeenth century. In India, however, Āryabhaṭa I
(499), Bhāskara I (522), Lalla (c. 598) and Brahmagupta (628)
all use the place value numerals. There is no trace of any other
system in their works. Perhaps in this country we had the place
value system as early as 200 B.C. if not earlier. The use of a
symbol for zero is found in Piṅgala's Chandaḥ Sūtra (perhaps of
200 B.C.). In literature, we have an indication of the place
value from about 100 B.C. and later in the Purāṇas from the
second to the fourth century A.D. The Bakhāśalī Manuscript
(perhaps of 200 A.D.) uses the place-value notations. The
earliest use of the place value principle with the letter numerals

1. ग्रहयोगोत्र ग्रहयुतिरार्यांत्रिशतियुताष्टसप्तत्या ।
   अध्याब्दैरभिभू लिकर्म वोच्चैर्विना भङ्गै ॥ —BrSpSi. X. 66.
2. गुरुणा न धूलिकर्म प्रतिकंचुककारिणे प्रदातव्यम् ।
   दत्तं सुकृतप्रणाशं कुरुते प्रतिकंचुकं यस्य ॥ —BrSpSi. X. 67

OPERATIONS AND DETERMINATIONS IN PĀṬĪGAṆITA 155
is found in the works of Bhāskara I about the beginning of the
sixth century A.D. Thus for 3179, the expressive words are
_Navādrirūpāgni_¹ (nava 9, adri 7. rupa 1 and agni 3). Similarly
in the Brāhmasphuṭasiddhānta, for a large number like 2296828522,
the expressive terms are DVIYAMAŚARĀṢṬAPAKṢAVASŪRA-
SANAVADVIYAMĀḤ(Dviyama two twos 22, Śara 5, aṣṭa 8, pakṣa
2, vasū 8, rasa 6. nava 9, dviyamāḥ 22).² Such usages are to be
found in all works, which clearly state the place value concept
was popular as a routine. From India, this system reached Arabia.
During the reign of the Khalif Al-Mansur (753-774 A.D.) there
came embassies from Sindh to Baghdad, and among them were
scholars, who brought along with them several works on mathe-
matics including the Brāhmasphuṭasiddhānta and the Khaṇḍa-
khādyaka of Brahmagupta: With the help of these scholars, Al-
fazari, perhaps also Yakub ibn Tarik, translated them into Arabic.
Both works were largely used and exercised great influence on
Arab mathematics. It was on that occasion that the Arabs first
became acquainted with a scientific system of astronomy. It is
acceptable to all writers on the subject that it was at that time
that the Hindu numerals were first definitely introduced amongst
the Arabs. Arabs at first adopted the ghobar form of num-
erals which they had already obtained (but without zero) from
the Alexandrians or from the Syrians. This they continued for
about two centuries, but since they were not suited to their right-
to-left script, they gave them up and adopted the more convenient
ones. For a detailed discussion on how numerals went to the
west from India and spread in Europe one is referred to this dis-
cussion in the History of Hindu Mathematics. Part I by Datta
and Singh (1935, Single volume Edition, 1962, pp. 83-104).
It is remarkable that Brahmagupta's works like the Brāhma-
sphuṭasiddhānta and the Khaṇḍakhādyaka became instrumental in
the spread of the place-value notation in the neighouring coun-
tries of the Middle East, and from their this system spread into
Europe.
Operations and Determinations in Pāṭīgaṇita
The word Pātīgaṇita is a compound formed from the words
pāṭī, meaning 'board', and gaṇita, meaning 'science of calculation',

1. MBh. 1. 4;
2. BrSpSi 1-16.

156 BRAHMAGUPTA AND ARITHMETIC
hence it means the science of calculation which requires the uses
of writing material (the board). The word pāṭī is not Sanskrit
(it originated in the non-Sanskrit literature in India); the oldest
term in Sanskrit for the board is Phalaka or paṭṭa. However this
term got currency in the Sanskrit literature also about the
beginning of the seventh century. Brahmagupta does not use
the term pāṭīgaṇita: he favours the use of the term dhūlikarma or
writing figures on dust spread on a board or on the ground. The
word pāṭīgaṇita was translated into Arabic as ilm-hisab-al-takht
(calculation on board) and the word dhūlikarma as hisab-al-ghobār
(calculation on dust).
Brahmagupta, in the very first verse in the Chapter XII
(Gaṇitādhyāya) refers to twenty operations (parikarma) and
eight determinations :
He who distinctly and severally knows the twenty logistics,
addition etc.. and the eight determinations (vyavahāra)
inculding (measurement by) shadow is a gaṇaka (mathe-
matician).¹
The commentators have given the list of these logistics
(parikarma) and determinations (vyavahāra) as follows;
(A) Parikarma or logistics

1. Saṁkalitam (addition)
2. Vyavakalitam (subtraction)
3. Gaṇanam (multiplication)
4. Bhāgahāraḥ (division)
5. Vargaḥ (square)
6. Vargamūlam (square-root)
7. Ghanaḥ (cube)
8. Ghanamūlam (cube root)
9. 13. Five standard forms of fractions (Pañca-jāti)
10. Trairāśikam (the rule of three)
11. Vysta-trairāśikam (the inverse rule of three)
12. Pañca-rāśikam (the rule of five)
13. Sapta-rāśikam (the rule of seven)
14. Nava-rāśikam (the rule of n ine)

1. परिकर्म विंशतिं यः संकलिताद्यं पृथक् विजानाति ।
   अष्टौ च व्यवहारान् छायान्तान् भवति गणकः सः ॥ —BrSpSi. XII. 1

MULTIPLICATION 157
19. Ekādaśa-rāśikam (the rule of eleven)
20. Bhāṇḍa-pratibhāṇḍam (barter and exchange)
(B) Vyavahāra or determinations

1. Miśrakaḥ (mixture)
2. Śreḍhī (progession or series)
3. Kṣetram (plane figures)
4. Khātam (excavation)
5. Citiḥ (stock)
6. . Krākacikaḥ (saw)
7. Rāśiḥ (mound)
8. Chāyā (shadow)
   Of the operations enlisted here, the first eight have been
   considered fundamental by later writers as Mahāvīra. The opera-
   tions of duplation and mediation (doubling and halving) were
   considered fundamental by Arabs, Greeks and Egyptians; since
   they were not familiar with the place-value system.
   Mathematics in this country developed as an aid to astro-
   nomy, and therefore, for the first time we find Āryabhaṭa(499AD.)
   in his Āryabhaṭīya describing as a special section (Gaṇitapāda).
   Brahmagupta (628 A.D.) also followed Āryabhaṭa in this respect
   and gave the science of calculation (gaṇita) a special place in his
   treatise on astronomy. The Siddhānta treatises, earlier than those
   of Āryabhaṭa and Brahmagupta do not contain a chapter exclu-
   sively devoted to gaṇita (the Sūrya-Siddhānta and the Siddhāntas
   of Vasiṣṭha, Pitāmaha and Romaka are thus without gaṇita
   chapetrs). Later on Bhāskara I and Lalla also did not include
   gaṇita as a section or chapter in their treatises. It is said, howe-
   ver, that Lalla wrote a separate treatise on Pāṭīgaṇita.
   It may further be remarked here that Āryabhaṭa I gives
   the rules for finding the square and cube-roots only whilst Bra-
   hmagupta gives the cube-root rule only (BrSPSi. XII. 7).
   Multiplication
   Undoubtedly the common Indian name ‘multiplication’ is
   ‘guṇana’, this term occurs in the Vedic literature also. The other
   terms for this logistics are hanana, vadha, kṣaya etc., which all
   mean ‘killing’ or ‘destroying.’ The synonyms of ‘hanana’ (killing)
   for multiplication have been used by Āryabhaṭa I (499). Brahma-
   gupta (628), Śrīdhara (c.750) and later writers, and these terms

158 BRAHMAGUPTA AND ARITHMETIC
also occur in the Bakhaśālī Manuscript.
Āryabhaṭa I does not mention the everyday methods of
multiplication in his Āryabhaṭīya probably because they were
too elementary to be included in a Siddhānta work. Brahma-
gupta, however, in a supplement to the section on mathematics
in his Siddhānta, gives the names of some methods with very
brief descriptions of the processes:—
The multiplicand repeated, as in gomūtrikā as often as
there are digits in the multiplier, is severally multiplied
by them and (the results) added according to places;
this gives the product. Or the multiplicand is repeated
as many times as there are component parts in the
multiplier.¹
(the word bheda occurring in the verse has . been translated
as "integrant portions" by Colebrooke p. 319. Again by the term
bheda are meant portions which added together make the whole,
or aliquot parts which multiplied together make the entire
quantity.
The multiplicand is multiplied by the sum or the differ-
ence of the multiplier and an assumed quantity and,
from the result the product of the assumed quantity
and the multiplicand is subtracted or added.²
(Colebrooke thinks that this is a method to obtain the true
product when the multiplier has been taken to be too great or
too small by mistake.³ Datta and Singh think, however, that
this is not correct.⁴
Thus Brahmagupta mentions four methods of multiplica-
tion: (i) gomūtrikā, (ii) khaṇḍa, (iii) bheda, and (iv) iṣṭa. The com-
mon and the well known method of kapāṭa-sandhi has been omitted
by him.

1. गुणकारखण्डतुल्यो गुण्यो गोमूत्रिकाकृतो गुणितः ।
   सहितः प्रत्युत्पन्नो गुणकारक्रमेदतुल्यो वा ॥ — BrSpSi XII. 55
2. गुण्यो राशिर्गुणकारराशिनेष्टाधिकोनकेन गुणः ।
   गुण्योष्टवधो न युतो गुण्येऽभ्यधिकोनके कार्यः ॥ — BrSpSi. XII. 56
3. Colebrooke, T. H., Hindu Algebra, p. 320.
4. Datta, B. and Singh, A. N., History of Hindu Mathematics Pt. I
   (Arithmetic), p. 135 (1962).

MULTIPLICATION 159
(i) Gomūtrika-method or zig-zag method. The word gomū-
trikā means "similar to the course of cow's urine", hence
"zigzag". This method in all essentials is the same as the sthāna-
khaṇḍa method. The following illustration is based on the
commentary of Pṛthūdaka Svāmī :
Example : To multiply 1223 by 235.
The numbers are written thus :
2 1223
3 1223
5 1223
The first line of figures is t hen multiplied by 2, the process
beginning at units place, thus : 2×3=6; 3 is rubbed out and 6
substituted in its place, and so on. After all the horizontal lines
have been multiplied by the corresponding numbers on the left
in the vertical line, the numbers on the pāṭī stand thus :
2446
3669
6115

287405
after being added together as in the present method.
The sthāna-khaṇḍa and the gomūtrikā methods resemble
modern plan of multiplication most closely.
(ii) Khaṇḍa Method · or Parts Multiplicatian Method :
Since the days of Brahmagupta, this method of multiplication
also became very popular. We have two methods under this
head :
(i) The multiplier is broken up into two or more parts
whose sum is equal to it. The multiplicand is then multiplied
severally by these and the results added.
To take an example :
13×158=(6+7)×158=(6×158)+(7×158)
=948+1106
=2054
(ii) The multiplier is broken up into two more aliquot
parts. The multiplicand is then multiplied by one of these, the
resulting product by the second and so on till all the parts are
exhausted. The ultimate product is the result.

160 BRAHMAGUPTA AND ARITHMETIC
Thus for example :
96 × 237 = (4 × 4 × 6) × 237
= (4 × 237) × 4 × 6 = 948 × 4 × 6
= (4 × 948) × 6 = 3792 × 6
= 22752
These methods of multiplication are found among the
Arabs and the Italians, having obtained from people of India.
They were known as the “Scapezzo” and “Repiego” methods
respectively amongst Italians.
(iii) Iṣṭa-guṇana Method or the Algebraic Method.
We have already quoted the relevant verse from the Brāh-
masphuṭa-siddhānta in this connection; (XII. 56) :
The multiplicand is multiplied by the sum or the
difference of the multiplied and an assumed quantity
and from the result the product of the assvmed quan-
tity and the multiplicand is subtracted or added.¹
[L

SQUARE 161
bhājaka, bhāgahāra or simply hara; quotient is known as labdhi or
labdha (or "what is obtained").
India never regarded this operation as a difficult one; in
Europe, this operation was regarded as a tedious one till the
15th century or so. Division was such a common operation that
Āryabhaṭa did not regard it as worth being included in his
treatise. But since he has given the methods of extracting
square-roots and cube-roots, which obviously depend on division,
we conclude that the method of division was known to him.
Most Siddhānta writers have followed Āryabhaṭa I in omitting
this operation from their texts, this being regarded too elementary
to be included. Brahmagupta does not give details of this
operation. The later treatises on Arithmetic as Śrīdhara's
Triśatikā and the Pāṭīgaṇita (I.20) and Āryabhaṭa II (c.950 A.D.)
have given the details of this operation.
Square
The Sanakrit term for square is varga or kṛti (varga lite-
rally means "rows" or "troops" of similar things). In mathema-
tics, it usually means the square power and also the square figure
or its area. Thus we find in the Āryabhaṭīya :
A square figure of four equal sides (and the number
representing its area) are called varga. The product of
the two equal quantities is also varga¹.
The term kṛti means "doing", "making" or "action". It
carries with it the idea of specific performance probably the gra-
phical representation.
For the first time we have a definite rule for squaring in the
writings of Brahmagupta. But it does not mean that prior to him
it was not known. It must have been known to Āryabhaṭa I
since he has given the square-root method.
Brahmagupta gives his method of squaring briefly as
follows :
Combining the product, twice the digit in the less
(lowest) place into the several others (digits) with its
(i.e. of the digit in the lowest place) square (repeatedly)
gives the square.²

1. वर्गस्समचतुरश्रः फलञ्च सदृशद्वयस्य संवर्गः। —Ārya. II. 3.
2. राशेरूनं द्विगुणं बहुतरगुणमूनकृतियुतं वर्गः। —BrSpSi. XII. 63.

162 BRAHMAGUPTA AND ARITHMETIC
The method has been more clearly enunciated by Mahāvīra
(850 A.D.) in the Gaṇitasārasaṁgraha :
Having squared the last (digit), multiply the rest by
the digits by twice the last, (which) is moved forward
(by one place). Then moving the remaining digits con-
tinue the same operation (process), This gives the
square.¹
Brahmagupta's method of squaring is shown by the follow-
ing example :
To square 125.
The number is written down
125
The square of the digit in the last place, i. e., 5²=25 is set
over it thus :
25
125
Then, 2 x 5=10 is placed below the other digits, and 5 is
rubbed out, thus :
25
12
10
Multiplying by 10 the rest of the digits, i.e., 12 and setting
the product over them (the digits), we have.
1225
12
10
Then rubbing out 10 which is not required and moving the
rest of the digits, i. e. 12 we, have
1225
12
Thus one round of operations is completed.
Again as before, setting the square of 2 above it and 2×2=4
below 1. we have
1625
1
4

1. GSS. P. 12.

C U B E 163
Multiplying the remaining digit 1 by 4, and setting the
product above it, we have
5625
1
Then moving the remaining digit 1, we obtain
5625
1
Thus the second round of operations is completed.
Next setting the square of 1 above it the process is comple-
ted for there are no remaining figures, and the result stands thus :
15625
Algebraic Method of Squaring
Brahmagupta in his Brāhmasphuṭasiddhānta gives a minor
method of squaring thus :
The product of the sum and the difference of the
number (to be squared) and an assumed number plus
the square of the assumed number give square¹.
This may be represented by the following identity :
n²=(n—a) (n+a)+a²
This identity has been used for squaring by most of the
Indian mathematicians. Thus
15²=(15—5) (15+5)+5²=225
We are not giving here other identities which have been
used by latter mathematicians of India in getting the squares
of numbers; for example, when Mahāvīra says :
The sum of the squares of the two or more portions of
the number together with their products each with the
others multiplied by two gives the square² :
he obviously refers to the identity
(a+b+c............)²=a²+b²+c²+......+2ab+......
Cube
The Sanskrit term for cube is ghana. It when used in
the geometrical sense also means the solid cube. In the arithmeti-
cal sense, it means the continued product of the same number
taken three times. Thus we have the definition in the Ārya-

1. राशेरिष्टयुतोनाद्वधः कृतिर्वेष्टकृतियुक्तः । — BrSpSi. XII. 63
2. GSS. p. 13.

164 BRAHMAGUPTA AND ARITHMETIC
bhaṭīya :
The continued product of three equals and also the
solid having twelve (equal) edges are called ghana.¹
A method of cubing applicable to numbers written in the
decimal place-value notation, has been in use in this country
from before the 5th century A.D. Āryabhaṭa I (499 A.D.) had
the familiarity with this method; he, however, does not give the
method of cubing in his treatise, though he describes the inverse
process of extracting the cube-root.
Brahmagupta gives the method of cubing in the following
verse :
Set down the cube of the last (antya); then place at the
next place from it, thrice the square of the last multi-
plied by the succeeding; then place at the next place
thrice the square of the succeeding multiplied by the
last, and (at the next place) the cube of the succeeding.
This gives the cube.²
The rule may be illustrated by an example.
Example : To cube 1357.
The given number has four places, i.e., four portions. First
we take the last digit 1 and the succeeding digit 3, i.e. 13 and
apply the method of cubing thus :
(i) Cube of the last (1³) = 1
(ii) Thrice the square of the
last (3.1²) multiplied by
the succeeding (3) gives
(3.3.1²) = 9 (placing at the
next place)
(iii) Thrice the square of the
succeeding, multiplied by
the last gives (3.3².1) = 27 (placing at the
next place)
(iv) Cube of the succeeding (3³) = 27 (placing at the
next place)
————————————————————————————————————————————————
Thus 13³ is the sum 2197

1. सदृशत्रयसंवर्गो घनस्तथा द्वादशाश्रयस्यात् ।। — Ārya, II. 3.
2. स्थाप्योऽन्त्यघनोऽन्त्यकृतिस्त्रिगुणोत्तरसङ्गुणा च तत्प्रथमात् ।
   उत्तरकृतिरन्त्यगुणा त्रिगुणा चोत्तर घनश्च घनः । — BrSPSi. XII, 6.

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164 BRAHMAGUPTA AND ARITHMETIC
bhaṭīya :
The continued product of three equals and also the
solid having twelve (equal) edges are called ghana.¹
A method of cubing applicable to numbers written in the
decimal place-value notation, has been in use in this country
from before the 5th century A.D. Āryabhaṭa I (499 A.D.) had
the familiarity with this method; he, however, does not give the
method of cubing in his treatise, though he describes the inverse
process of extracting the cube-root.
Brahmagupta gives the method of cubing in the following
verse :
Set down the cube of the last (antya); then place at the
next place from it, thrice the square of the last multi-
plied by the succeeding; then place at the next place
thrice the square of the succeeding multiplied by the
last, and (at the next place) the cube of the succeeding.
This gives the cube.²
The rule may be illustrated by an example.
Example : To cube 1357.
The given number has four places, i.e., four portions. First
we take the last digit 1 and the succeeding digit 3, i.e. 13 and
apply the method of cubing thus :
(i) Cube of the last (1³) = 1
(ii) Thrice the square of the
last (3.1²) multiplied by
the succeeding (3) gives
(3.3.1²) = 9 (placing at the
next place)
(iii) Thrice the square of the
succeeding, multiplied by
the last gives (3.3².1) = 27 (placing at the
next place)
(iv) Cube of the succeeding (3³) = 27 (placing at the
next place)
——————
Thus 13³ is the sum 2197

1. सदृशत्रयसंवर्गो घनस्तथा द्वादशाश्रस्स्यात् ॥ — Ārya, II. 3.
2. स्थाप्योऽन्त्यघनोऽन्त्यकृतिस्त्रिगुणोत्तरसंगुणा च तत्प्रथमात् ।
   उत्तरकृतिरन्त्यगुणा त्रिगुणा चोत्तर घनश्च घनः । — BrSPSi. XII, 6.

C U B E 165
After this we take the next figure, 5, i.e., the number 135,
and in this consider 13 as the last and 5 as the succeeding. Then
the method proceeds thus :
(i) The cube of the last
(13³) as already obtained = 2197
(ii) Thrice the square of the
last multiplied by the
succeeding, i.e. 3.13².5 = 2535 (placing at the
next place)
(iii) Thrice the square of the
succeeding multiplied
by the last, i.e. 3.5².13 = 975 (placing at the
next place)
(iv) Cube of the succeeding,
i.e. 5³ = 125 (placing at the
next place)
Thus 135³ is the sum 2460375
Now the remaining figure 7 is taken, so that the number is
1357, of which 135 is the last and 7 the succeeding. The method
proceeds thus :
(i) Cube of the last, i.e.
(135³) as already
obtained = 2460375
(ii) Thrice the square of
the last into the succee-
ding, i.e. 3. (135)². 7 = 382725 (placing at the
next place)
(iii) Thrice the square of
the succeeding into the
last, i.e. 3.7². 135 = 19845 (placing at the
next place)
(iv) Cube of the succeeding
i.e. 7³ = 343 (placing at the
next place)
————————
Thus (1357)³ is the sum 2498846293
Evidently these methods of cubing are based on the identity:
(a+b)³=a³+3a²b+3ab²+b³
and keeping in mind the place values of numerals in a given