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

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

166 BRAHMAGUPTA AND ARITHMETIC
number (this accounts for keeping the results of each of the four
operations at the next place).
Square-Root
Indian synonyms for square-root are vargamūla or pada
of a kṛti. The word mūla means the "root" of a tree, which may
also mean the "foot" or the lowest part or bottom of a thing and
hence "pada" or foot also became a synonym of root. Brahma-
Gupta defines square-root as follows :
The pada (root) of a kṛti (square) is that of which it is
the square.¹
While the word mūla for root is the oldest in Indian litera-
ture (it occurs in Anuyogadvāra-sūtra, c. 100 B.C.), the word
pada for root probably for the first time occurs in the writings
of Brahmagupta. The term mūla was borrowed by the Arabs who
translated it by jadhr, meaning "basis of square". The Latin term
radix also is a translation of the term mūla. In the Śulba litera-
ture and in the Prākṛta texts, we find a term karaṇī for square-
root. In geometry, this term karaṇī means a "side", In later days,
the term karaṇī was reserved for surds, i.e. a square-root which
cannot be exactly evaluated, but which may be represented by a
line.
We would like to quote here a rule for determining square-
root of numbers from the Āryabhaṭīya :
Always divide the even place by twice the square-root
(upon the preceding odd place); after having subtracted
from the odd place the square (of the quotient), the
quotient put down at the next place (in the line of the
root) gives the root².
As an illustration, we shall proceed to find the square-root
of 18225.
The odd and even places are marked out by vertical (I) and
horizontal (—) lines : The other steps are as follows :

1. पदं कृतिय॑त् तत् BrSpSi. XVIII. 35
2. भागं हरेदवर्गान्नित्यं द्विगुणेन वर्गमूलेन ।
   वर्गाद्वर्गे शुद्धे लब्धं स्थानान्तरे मूलम् ॥ — Ārya, II. 4.

CUBEROOT 167
| - | - |
1 8 2 2 5
Subtract square 1 root = 1
Divide by twice ———
the root 2) 8 (3 placing quotient at the
6 next place, the root=13
———
22
Subtract square of
quotient 9
Divide by twice 26)132(5 placing quotient at the
the root 130 next place, the root=135
Subtract square 25
of the quotient 25
The process ends. The square-root of 18225 is thus 135.
It has been stated by Kaye, that Āryabhaṭa’s method of
finding out the square-root is algebraic in character, and that it
resembles the method given by Theon of Alexandria. Ārya-
bhaṭa’s method is purely arithmetic and not algebraic is the
view of Datta and Singh who do not agree with Kaye on this
point.
Cube Root
The Sanskrit term for cube-root is ghanamūla or ghanapada.
The first mention of the operation of cube-root is found in the
Āryabhaṭīya of Āryabhaṭa I (499 A.D.), though the operation
is given in only a concise form :
Divide the second aghana place by thrice the square
of the cube-root; subtract from the first aghana place
the square of the quotient multiplied by thrice the
preceding (cube-root); and (subtract) the cube (of the
quotient) from the ghana place; (the quotient put
down) at the next place (in the line of the root) gives
(the root).¹
As has been explained by all the commentators on the
Āryabhaṭīya, the units place is ghana; the tens place is first
aghana, the hundreds place is the second aghana, the thousands
place is ghana, the ten thousands place is first aghana, the hun-

1. अघनाद् भजेद् द्वितीयात् त्रिगुणेन घनस्य मूलवर्गेण।
   वर्गस्त्रिपूर्व गुणितश्शोध्यः प्रथमाद् घनश्च घनात् ॥ —Ārya. II. 5

168 BRAHMAGUPTA AND ARITHMETIC
dred-thousands place is second aghana, and so on. Thus to find
out the cube-root, one has to mark out the ghana, first aghana
and second aghana places, then the process of finding out the
cube-root begins with the subtraction of the greatest cube num-
ber from the figures up to the last ghana place. Though this has
not been explicitly mentioned in the rule, the commentators
say that it is implied in the expression ghanasya mūla-vargeṇa
etc. ("by the square of the cube-root etc.")
We are reproducing here an illustration given by Datta
and Singh.
Example. Find the cube-root of 1953125.
The places are divided into groups of three by marking
them as below [ghana ( | ) first aghana (—) and second aghana
(--)]:
| — — | — — |
1 9 5 3 1 2 5
Subtract cube 1 ... ... ... ... ... (c) Root=1
Divide by thrice ——————————————
square of root,
i.e. 3.1² 3)9(2 ... (a) Placing quotient
Subtract square 6 after the root 1
of quotient mul- —— gives the root 12
tiplied by thrice 35
the previous root, 12 ... (b)
i.e. 2².3.1 ——
Subtract cube of 233
quotient, i.e. 2³ 8 ... (c)
Divide by thrice
square of the root,
i.e. 3.12² 432)2251(5 ... (a) Placing quotient
Subtract square 2160 after the root
quotient multiplied ———— 12 gives the
by thrice the pre- 912 root 125
vious root, i. e.
5².3.12 900
————
Subtract cube of 125 ... (b)
quotient, i.e. 5³ 125 ... (c)
————
Thus the cube-root=125.
From the details given, it would be clear that the present

CUBE ROOT 169
method of extracting the cube-root is almost a contraction of the
method first given by Āryabhaṭa I (499 A.D.)
The method of Āryabhaṭa has been invariably followed by
Indian mathematicians. Brahmagupta in his Brāhmasphuṭa-
siddhānta repeats the method in the following words :
The divisor for the second aghana place is thrice the
square of the cube-root; the square of the quotient
multiplied by three and the preceeding (root) must be
subtracted from the next (aghana place to the right).
and the cube (of the quotient) from the ghana place
(the procedure repeated gives) the root.¹
Śrīdhara and Āryabhaṭa II have further improved on the
method of extracting cube-root proposed by Āryabhaṭa I and
followed by Brahmagupta. Rule for finding the cube-root as
given by Śrīdhara in his Pāṭīgaṇita is as follows :
(Divide the digits beginning with the units' place into
periods of) one ghana-pada (one "cube" place) and two
aghana-padas (two "non-cube" places). Then subtrac-
ting the (greatest possible) cube from the (last) ghana-
pada and placing the (cube) root underneath the
third place (to the right of the last ghana-pada),
divide out the remainder up to one place less (than
that occupied by the cube-root) by thrice the square
of the cube-root, which, is not destroyed. Setting
down the quotient (obtained from division) in the
line (of the cube-root), (and designating the quotient
as the 'first' (ādima) and the cube-root as the 'last'
(antya), subtract the square of that quotient, as multi-
plied by thrice the 'last' (antya) from one place less
than that occupied by the quotient (uparima-rāśi) as
before, and the cube of the 'first' (ādima) from its own
place.
(The number now standing in the line of cube-root is
the cube-root of the given number up to its last-but
one ghana-pada (cube place) from the left).
Again apply the rule, "(placing cube-root) under the
third place" etc. (provided there be more than two
ghana-padas (cube places) in the given number; and

1. छेदो घनाद् द्वितीयाद् घनमूलकृतेस्त्रिसंगुणाप्ताप्तकृतिः ।
   शोध्या त्रिपूर्वगुणिता प्रथमाद् घनतो घनो मूलम् ॥ —BrSpSi. XII. 7

170 BRAHMAGUPTA AND ARITHMETIC
continue the process till all ghana-padas (cube-places)
are exhausted). This will give the (cube) root (of the
given number).¹
K.S. Shukla in his translation and commentary of this book
has given the illustration of extracting cube - root as
follows :
Example :- To find the cube root of 277167808.
Let us indicate ghana-padas or 'cube' places by "c" and
aghana-padas or non-cube places as "n" :
n n c n n c n n c
2 7 7 1 6 7 8 0 8
Subtract the greatest possible cube (i.e. 6³ or 216) from
the last 'cube' place (i.e. from 277) and place the cube-
root (i.e. 6) underneath the third place to the right of
the last 'cube' place; thus we have
n n c n n c n n c
6 1 1 6 7 8 0 8 (remainder)
6 (line of cube-root)
Dividing out by thrice the square of the cube-root (i.e.
by 3.6² or 108) the remainder up to one place less
than that occupied by the cube-root (i.e. 611) and
setting down the quotient in the line of the cube-root
(to the right of the cube-root), we have
n n c n n c n n c
7 1 6 7 8 0 8 (remainder)
6 5 (line of cube-root)
Let now quotient 5 be called the 'first' (ādima) and
the cube-root 6 the 'last' (antya). Then subtracting
the square of the 'first' (ādima) as multiplied by thrice
the 'last' (antya) (i.e. 3×6×5² or 450) from one place
less than that occupied by the quotient (i.e. from 716),
we get

1. घनपदमघनपदे द्वे घन (पद) तोऽपास्य घनमदो मूलम् ।
   संयोज्य तृतीयपदस्याधस्तदनष्टवर्गेण ॥ २९ ॥
   एकस्थानोनतया शेषं त्रिगुणेन (सं) भजेत्तस्मात् ।
   लब्धं निवेश्य पङ्क्त्यां तद्वर्गं त्रिगुणमन्त्यहतम् ॥ ३० ॥
   जह्यादूपरिमराशेः प्राग्वद् घनमादिमस्य (च) स्वपदात् ।
   भूयस्तृतीय पदस्याध इत्यादिक विधिर्मूलम् ॥ ३१॥
   —Śrīdhara, Pāṭīgaṇita, 29-31

CUBEROOT 171
n n c n n c n n c
2 6 6 7 8 0 8 (remainder)
6 5 (line of cube-root)
And subtracting the cube of the 'first' (ādima) (i.e. 5³
or 125) from its own place (i.e. from 2667), we get
n n c n n c n n c
2 5 4 2 8 0 8 (remainder)
6 5 (line of cube-root)
One round of the operation is now over; and the num-
ber 65 standing in the line of the cube-root is the cube-
root of the given number (277167808) up to its last-but-
one 'cube' place ( ghana pada ) from the left (i.e. of
277167),
As there is one more 'cube' place (ghana-pada) on the
right, the process is repeated. Thus placing the cube-
root (i.e. 65) under the third place beginning with the
last-but-one 'cube' place (ghana-pada), we have
n n c n n c n n c
2 5 4 2 8 0 8 (remainder)
6 5 (line of cube-root)
Dividing out 25428 by 3.65² (=12675) as before, and
placing the quotient in the line of the cube-root, we
have
n n c n n c n n c
7 8 0 8 (remainder)
6 5 2 (line of cube-root)
Subtracting 3 × 65 × 2² (=780) we get.
n n c n n c n n c
8 (remainder)
6 5 2 (line of cube-root)
Finally subtracting 2³=8 from 8, we get
n n c n n c n n c
0 (remainder)
6 5 2 (line of cube-root)
The second round of operation is now over. There
being no more of ghana-pada ('cube' place) on the
right, the process ends. The quantity in the line of
cube root, viz., 652, is the cube-root of the given

172 BRAHMAGUPTA AND ARITHMETIC
number. The remainder being zero, the cube-root
is exact.
Fractions
The concept of fractions in India can be traced to very
early times. In the Ṛgveda,¹ we find such terms as one-half
(ardha) and three-fourths (tri-pāda). In a passage of the
Maitrāyaṇī Saṁhitā² are mentioned the fractions one-sixteenth
(kalā), one-twelfth (kuṣṭha), one-eighth (śapha) and one-fourth
(pāda). In the Śulba Sūtras³ we have not only a mention of
fractions, but they have been used in the statement and solution
of problems of geometric nature. Here in the Śulba, unit frac-
tions are denoted by the use of cardinal number with the term
bhāga or aṁśa; thus pañca-daśa-bhāga (literally “fifteen parts”) is
equivalent to one-fifteenth, sapta-bhāga (literally, “seven parts”)
is equivalent to one-seventh, and so on... The use of ordinal
numbers with the term bhāga or aṁśa is also quite common :
thus pañcama bhāga stands for one-fifth. The composite fractions
like tri-aṣṭama stands for three-eighths and dvi-saptama for two-
sevenths. In the Bakhshālī Manuscript, the term tryaṣṭa occurs
for 3/8 and 3⅜ is called trayastrayasta (three-three-eighths).
The Sanskrit term for fraction is bhinna (literally meaning
‘broken’). Obviously the European terms as fractio, fraction,
roupt, rotto or rocto are translations of the same term; they are
derived from the Latin fractus (frangere) or ruptus meaning
‘broken’. The Indian term bhinna has a few more connotations;
it stands for such numbers of the form :
(a/b ± c/d), (a/b of c/d), (a/b ± c/d of a/b) or (a ± b/c).
These forms were termed jāti’ i.e., ‘classes’, and the Indian
treatises contain special rules for their reduction to proper frac-
tions. Śrīdhara and Mahāvīra each enumerate six jātis, while our
author, Brahmagupta, gives only five (Bhāskara II gives only
four). The need for division of fractions in ‘classes’ arose out
of the lack of proper symbolism to indicate mathematical opera-
tions. (Datta and Singh Arithmetic, p. 188). The only operational
symbol in use was a dot, standing for the negative sign.

1. Ṛv. X, 90,4
2. Mait S, III, 7,7.
3. B. Datta, Śulba, pp. 212ff.

FRACTIONS 173
Reduction to lowest terms.—A non-mathematical work,
Tattvārthādhigama-Sūtra-Bhāṣya by Umāsvāti (c.150A.D.) casua-
lly mentions as follows in the context of a philosophic discourse:
Or, as when the expert mathematician, for the purpose
of simplifying operations, removes common factors
from the numerator and denominator of a fraction,
there is no change in the value of the fraction, so....¹
Reduction to common denominator. Whenever we have to
add or subtract fractions, we follow this reduction operation
to a common denominator. Brahmagupta gives the reduction
along with the similar processes :
By the multiplication of the numerator and denomi-
nator of each of the (fractional) quantities by other
denominators, the quantities are reduced to a common
denominator. In addition, the numerators are united,
In subtraction their difference is taken.²
Fractions in combination :—Since there was no proper
symbolism available to these early Indian mathematicians, they
divided combination of fractions into four classes :
Bhāga, prabhāga, bhāgapavāha and bhāga-bhāga.
(i) Bhāga has been mentioned by Brahmagupta (BrSpSi.
XII, 8) thus : ( a/b ± c/d ± e/f ± ........ ) usually written as
┌───┬───┬───┐ ┌───┬────┬────┐
│ a │ c │ e │ or │ a │ .c │ .e │
├───┼───┼───┤ ├───┼────┼────┤
│ b │ d │ f │ │ b │ d │ f │
└───┴───┴───┘ └───┴────┴────┘
where the dots denote subtraction.
(ii) Prabhāga : The form ( a/b of c/d of e/f ......... )
This is written as
┌───┬───┬───┐
│ a │ c │ e │
├───┼───┼───┤
│ b │ d │ f │
└───┴───┴───┘
(iii) Bhāgānubandha : The form ( a + b/c ) is written as
┌───┐
│ a │
├───┤
│ b │
├───┤
│ c │
└───┘

1. II, 52.
2. विपरीतच्छेदगुणा : राश्योश्छेदांशाः समच्छेदाः ।
   संकलितेंऽशा योज्या व्यवकलितेंऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2.