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

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.

174 BRAHMAGUPTA AND ARITHMETIC
and the form
p/q + r/s of p/q + t/u of (p/q + r/s of p/q) + .........
is written as
┌───┐
│ p │
│ q │
├───┤
│ r │
│ s │
├───┤
│ t │
│ u │
└───┘
(iv) Bhāgāpavāha, i.e., the form (a - b/c) is written as
┌───┐
│ a │
│ .b│
│ c │
└───┘
and the form p/q - r/s of p/q - t/u of (p/q - r/s of p/q) - ........
is written as
┌───┐
│ p │
│ q │
├───┤
│ .r│
│ s │
├───┤
│ .t│
│ u │
└───┘
(v) Bhāga-bhāga : The form
(a ÷ b/c) or (p/q ÷ r/s)
There does not appear to have been any notation for divi-
sion, such compounds being written as
┌───┐ ┌───┐
│ a │ │ p │
│ b │ or │ q │
│ c │ ├───┤
└───┘ │ r │
│ s │
└───┘
just as for bhāgānubandha. That division is to be performed was
known from the problem, e.g., 1 ÷ 1/6 was written as ṣaḍ-bhāga-
bhāga, i.e., "one-sixth bhāga-bhāga" or "one divided by one-
sixth". It is only in the Bakhshālī Manuscript that the term bha
is sometimes placed before or after the quantity affected.
(vi) Bhāga-mātṛ, i.e., combinations of forms enumerated
above. Mahāvīra, the author of the Gaṇitasārasaṃgraha (850

FRACTIONS 175
A.D.) gives twenty-six variations of this class. We shall illus-
trate it by the following example from Śrīdhara :
What is the result when half, one-fourth of one-fourth,
one divided by one-third, half-plus half of itself, and
one-third diminished by half of itself, are added
together ? (Triśatikā, p. 12).
A modern writer would have written it as :
½+(¼ of ¼)+(1÷⅓)+(½+½ of ½)(⅓—½ of ⅓)
In the old Indian notation, it is written as ·
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 4 | 1 | 2 | 3 |
| | | | 3 | 1 | .1 |
| | | | | 2 | 2 |
The defect of the notation is obvious: | 1 | 1 | can be read
| 4 | 4 |
also as ¼+¼ and | 1 | can also be read as 1 ⅓.
| 1 |
| 3 |
And therefore the original meaning is inferred from the
context or from the enunciation of the problem.
The rules for reduction of the first two classes (bhāga and
prabhāga) are those of addition or subtraction and multiplica-
tion. The rule for the reduction of the third (bhāgānubandha)
and fourth (bhāgāpavāha) classes are given by Brahmagupta in
the Brāhmasphuṭa-siddhānta thus :
The (upper) denominator is multiplied by the deno-
minator and the upper numerator by the same (deno-
minator) increased or diminished by its own numera-
tor.¹
“Numerator” is known as “aṁśa” and the “denominator”
as “cheda.”
We give here from Śrīdhara's Pāṭīgaṇita (about 900 A.D.,
according to K.S. Shukla, 750 A.D. according to Datta and
Singh) a rule for reducing a fraction of the bhāgānubandha class
(i.e., a whole number increased by a fraction or a fraction incre-
ased by a fraction itself) :

1. ऊर्ध्वां शारच्छेदगुखास्तृतीयाजातौ द्वयोः पृथक्परयोः ।
   छेदैश्छेदा गुणिताः स्वांशयुतोनैरुपरीमांशाः ॥ —BrSpSi. XII. 9.

176 BRAHMAGUPTA AND ARITHMETIC
In the bhāgānubandha class, the whole number (rūpa-
gaṇa) is multiplied by the denominator (of the frac-
tion) should be increased by the numerator (of the
fraction) or the upper denominator having been
multiplied by the lower denominator, the initial
numerator (i.e. the upper numerator) should be multi-
plied by the sum of the lower numerator and denomi-
nator.¹
(Pātīgaṇita, 39 cf. BrSpSi. XII. 9 (ii); GSS. (iii) 113
This means that
(i) a + b/c = (ac + b)/c
(ii) a/b + c/d of a/b (which was written by Indians in
the style
┌───┐
│ a │
│ b │
│ c │
│ d │
└───┘
is equal to a(d + c)/bd
Addition and Subtraction of Fractions
In the Brāhmasphuṭa-siddhānta, Brahmagupta gives the
rule for the addition and subtraction of fractions :
If the denominators (cheda) of fractions are different
then reduce these fractions to a common denomina-
tor. Now for the additions, unite the numerators and
take their difference in case of subtraction.²
Brahmagupta and Mahāvīra give the method under Bhāga-
jāti.
Multiplication
Brahmagupta says :
The product of the numerators divided by the pro-

1. भागानुबन्धजातौ रूपगणश्छेद सङ्गुणः सांशः ।
   अवरहरन्धोर्ध्वं हरेऽथोंशाद्युतहरघ्न आद्यंशः ॥ —Patīganita 39.
2. विपरीतच्छेदगुणाः राश्योश्छेदांशकाः समच्छेदाः ।
   संकलितेंऽशा योज्या व्यवकलितेऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2

DIVISION OF FRACTIONS 177
duct of the denominators is the (result of) multiplica-
tion of two or more fractions.¹
While all other writers give the rule in the same way as
Brahmagupta, Mahāvīra in the Gaṇitasārasaṃgraha refers to
cross reduction in order to shorten the work :
In the multiplication of fractions, the numerators are
to be multiplied by the numerators and the denomi-
nators by denominators, after carrying out the pro-
cess of cross reduction, if that be possible.²
Division of Fractions
The Āryabhaṭīya does not explicitly give the rule of divi-
sion, but under the Rule of Three, we have an indication of
this operation. The Rule of Three states the result as (f × i) / p,
where f stands for phala i.e. "fruit", i for icchā, i.e., demand or
requisition, and p for pramāṇa i.e. argument. When these
quantities are fractional, we get an expression of the form
(a/b × c/d) / (m/n)
for the evaluation of which Āryabhaṭa I states :
The multipliers and the divisor are multiplied by the
denominators of each other.
These quantities are written in the following way
┌───────┐
│ a m │
│ b n │
├───────┤
│ c │
│ d │
└───────┘
Transferring the denominators we have
┌───────┐
│ a m │
│ n b │
│ c d │
└───────┘
Performing multiplication, the result is anc / mbd. The above
interpretation of the obscure line in the Āryabhaṭīya is based

1. रूपाणिच्छेद गुण्यान्यंशयुतानि द्वयोर्बहूनां वा ।
   प्रत्युत्पन्नो भवति च्छेदवधेनोद्धृतोऽशवधः ॥ —BrSpSi. XII. 3
2. GSS. p. 25. (2)

178 BRAHMAGUPTA AND ARITHMETIC
on the commentaries of Sūryadēva and Bhāskara I (the commen-
tary of Parameśvara on this line is vague and misleading).
Sūryadeva in this connection says :
Here by the word guṇakāra is meant the multiplier
and multiplicand, i.e., the phala and icchā quantities
that are multiplied together. By Bhāgahāra is meant
the pramāṇa quantity. The denominators of the phala
and icchā are taken to the pramāṇa. The denominator
of the pramāṇa is taken with the phala and icchā. Then
multiplying these, i.e., (the numerators of) the phala
and icchā and this denominator, and dividing by (the
product of) the numbers standing with the pramāṇa
the result is the quotient of the fractions.
Brahmagupta gives the method of division as follows :
The denominator and numerator of the divisor having
been interchanged, the denominator of the dividend is
multiplied by the (new) numerator. Thus division of
*proper fractions is performed.*¹
Square and Square-Root of Fractions
Brahmagupta says as follows in this connection:—
The square of the numerator of a proper fraction divi-
ded by the square of the denominator gives the
square².
This rule of Brahmagupta has been followed by other
authors also. The rule regarding the square-root as given by
Brahmagupta is as follows :
The square-root of the numerator of a proper fraction divi-
ded by the square-root of the denominator gives the square-
root.³
The Rule of Three :
The Indian term in Sanskrit for the Rule of Three is Trai-
rāśika (literally, "three terms"). The term occurs in the Bakh-
śālī Manuscript also, and also in the Āryabhaṭīya, indicating the

1. परिवर्त्य भागहारच्छेदांशौ छेद संगुणच्छेदः ।
   अंशोंशगुणो भाज्यस्य भागहारः सवर्णितयोः ॥ —BrSpSi: XII. 4
2. संवर्गितांशवर्गश्छेदकृतिविभाजितो भवति वर्गः । —BrSpSi. XII. 5 (1)
3. संवर्गितांशमूलं छेदपदेनोद्धृतं मूलम् । —BrSpSi. XII. 5 (2)

THE RULE OF THREE 179
antiquity of the term. Bhāskara in his commentary of the
Āryabhaṭīya gives a justification of the use of this term for the
Rule of Three thus :
Here three quantities are needed (in the statement and
calculation) so the method is called trairāśika (meaning
thereby the "rule of three terms").
The problem of the Rule of Three has the form :
If p (pramāṇa) yields f (phala), what will i (icchā)
yield ?
Āryabhaṭa II (the author of the Mahāsiddhānta, 950 A.D.)
uses the terms māna, vinimaya and icchā, instead of pramāṇa,
phala and icchā respectively. It has also been pointed out by
several authors that the first and third terms are similar, i.e., of
the same denomination.
We shall give here the Rule of Three as given by Āryabhaṭa
I and Brahmagupta :
In the Rule of Three, the phala ("fruit"), being multi-
plied by the icchā ("requisition") is divided by the
pramāṇa ("argument"). The quotient is the fruit
corresponding to the icchā The denominators of one
being multiplied with the other give the multiplier
(i.e. numerator) and the divisor (i.e. denominator).¹
In the Rule of Three pramāṇa ("argument"), phala
("fruit") and icchā ("requisition") are the (given)
terms; the first and the last terms must be similar. The
icchā multiplied by the phala and divided by the
pramāṇa gives the fruit (of the demand).²
Śrīdhara also gives the Rule of Three almost in the same
words. Bhāskara II, Nārāyaṇa and others follow Brahmagupta
and Śrīdhara in the Trairāśika operation. Śrīdhara in his Pāṭīga-
nita says :

1. त्रैराशिकफलराशिं तमथेच्छाराशिनाहतं कृत्वा ।
   लब्धं प्रमाणभजितं तस्मादिच्छाफलमिदं स्यात् ।।
   छेदाः परस्परं हता भवन्ति गुणकार भागहाराणां ।
   छेदगुणं सच्छेदं परस्परं तत्सवर्णत्वम् ।। —Arya. II 26-27.
2. त्रैराशिके प्रमाणं फलमिच्छाद्यन्तयोः सदृशराशी ।
   इच्छाफलेन गुणिता प्रमाणभक्ता फलं भवति ।। —BrSpSi XII. 10

180 BRAHMAGUPTA AND ARITHMETIC
In (solving problems on) the Rule of Three, the argu-
ment (pramāṇa) and the requisition (icchā), which are
of the same denomination, should be set down in
the first and last places; the fruit (phala), which is of
a different denomination, should be set down in the
middle, (this having been done) that (middle quantity
multiplied by the last quantity should be divided by
the first quantity.¹
We shall illustrate the Rule of Three by an example from
the Pāṭīgaṇita (Example 25) :
Example, If 1 pala and 1 karṣa of sandalwood are obtai-
ned for ten and a half paṇas, then for how much will
nine palas and one karṣa (of sandalwood) be obtai-
ned ?²
Here in this Example.
argument = 1 pala and 1 karṣa = 1¼ or 5/4 palas; fruit =
10½ or 21/2 paṇas;
and requisition = 9 palas and 1 karṣa = 9¼ or 37/4 palas.
According to the Rule we shall write them as :

1. आद्यन्तयोस्त्रिराशावभिन्नजाती प्रमाणमिच्छा च ।
   फलमथ विजातीयं तदन्त्यगुणमादिना विभजेत् ॥ —Pāṭīgaṇita 43.
2. चन्दनपलं सकर्षं सार्धैर्यदि लभ्यते पणैर्दशभिः ।
   तत्किं नु लभ्यन्ते पलानि नव कर्षयुक्तानि ॥ —Pāṭīgaṇita Ex. 25.

RULE OF COMPOUND PROPORTION 181
=4 purāṇa, 13 paṇas, 2 kākiṇīs and 16 varāṭakas. (One purāṇa
is equivalent to 16 paṇas; one paṇa is equivalent to 4 kākiṇīs,
and one kākiṇī is equivalent to 20 varāṭakas or cowries.
Inverse Rule of Three
This is known as vyasta-trairāśika (literally meaning "inverse
rule of three terms)". After having described the rule of three,
Brahmagupta proceeds to give an account of this inverse rule
of three :
Divide the phala with icchā and multiply by pramāṇa;
this gives the vyasta-trairāśika inverse rule of three¹.
Here pramāṇa is the argument also known as the first term
and, and phala is the fruit also known as the middle term and
icchā is known as requisition or the last term. As Bhāskara II
clearly states, this rule is applied where with the increase of the
icchā, the phala decreases or with its decrease the phala increases
(Līlāvatī).
Rule of Compound Proportion
Brahmagupta and other writers call the rule of compound
proportions as pañca-rāśika, sapta-rāśika etc., meaning the
rule of five terms, rule of seven terms etc. depending on the
number of terms involved the problems. These are
sometimes grouped under the general application of the "Rule of
Odd Terms". Āryabhaṭa I (499 A.D.) though actually gives the
rule of three appears to have been quite familiar with the rule
of compound proportion also. In fact the difference between
the rule of three and compound proportion is more or less arti-
ficial. This view was expressed by Bhāskara I (525 A.D.) in his
commentary on the Āryabhaṭīya :
Here Ācārya Āryabhaṭa has described the Rule of
Three only. How the well-known Rules of Five etc. are
to be obtained ? I say thus : The Ācārya has described
only the fundamentals of anupāta (proportion). All
others such as the Rule of Five etc. follow from that
fundamental rule of proportion. How ? The Rule of
Five etc. consist of combinations of the Rule of Three.
......In the Rule of Five, there are two Rules of

1. व्यस्त त्रैराशिक फलमिच्छा भक्तः प्रमाण फलघातः ।
   त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥ —BrSpSi XII. 11

182 BRAHMAGUPTA AND ARITHMETIC
Three, in the Rule of Seven three Rules of Three, and
so on. This I shall point out in the examples.
Brahmagupta gives the following rule relating to the solu-
tion of problems in compound proportion :
In the case of odd terms beginning with three terms
up to eleven, the result is obtained by transposing
the fruits of both sides, from one side to the other,
and then dividing the product of the larger set of
terms by the product of the smaller set. In all the
fractions, the transposition of denominators, in like
manner, takes place on both sides.¹
This may be illustrated by taking an example from the
commentary of Pṛthūdaka Svāmī on the Brāhmasphuṭasid-
dhānta :
Example —If there is an increase of 10 in 3 months on
100 (niṣkas), what would be the increase on
60 (niṣkas) in 5 months.
Here the Pramāṇa pakṣa (the first set of terms) is
100 niṣkas, 3 months, 10 niṣkas (phala)
The second set or the icchā pakṣa is
60 niṣkas, 5 months, x niṣkas
The terms are written in compartments as below :
| 100 | 60 |
| 3 | 5 |
| 10 | 0 |
In the above 10 (written lowest) is the fruit of the first side
(pramāṇa pakṣa), and there is no fruit on the second side or the
iccā pakṣa. Interchanging the fruits we get
| 100 | 60 |
| 3 | 5 |
| 0 | 10 |
The larger set of terms is on the second side (icchā pakṣa).
The product of the numbers is 3,000. The product of the

1. व्यस्तं त्रैराशिक फलमिच्छा भक्तः प्रमाणफलघातः ।
   त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥
   फलसंक्रममुभयतो बहुराशि वधोऽल्पवधहृतो ज्ञेयम् ।
   सकलेष्वेवं भिन्नेष्वथवैतच्छेदसंक्रमकम् ॥
   —BrSpSi: XII. 11-12.

RULE OF THREE AS A PARTICULAR CASE 183
number on the side of the smaller set of terms is 300. Therefore
the required result is 3000 / 300 = 10.
Rule of Three as a Particular Case
According to Brahmagupta, the above method of "com-
pound proportion" may be applied to the Rule of Three. Taking
the example solved under the Rule of Three :
If one pala and one karṣa of sandal wood are obtained
for ten and a half paṇas, for how much will be obtain-
ed nine palas and one karṣa ?
(4 karṣas = 1 pala).
We shall represent them according to the Rule of Com-
pound Proportion as
Pramāṇa pakṣa : 1 pala, 1 karṣa, 10½ paṇa
or 5/4 pala , 21/2 paṇa,
Icchā pakṣa : 9 pala, 1 karṣa, x paṇa
or 37/4 pala , x paṇa
This we shall represent as
┌────┬────┐
│ 5 │ 37 │
│ 4 │ 4 │
├────┼────┤
│ 21 │ 0 │
│ 2 │ │
└────┴────┘
Transposing the fruits, we have
┌────┬────┐
│ 5 │ 37 │
│ 4 │ 4 │
├────┼────┤
│ 0 │ 21 │
│ │ 2 │
└────┴────┘
Transposing denominators
┌────┬────┐
│ 5 │ 37 │
│ 4 │ 4 │
├────┼────┤
│ 0 │ 21 │
│ 2 │ │
└────┴────┘
The product of numbers on the side of the larger set is
divided by the product of the numbers on the side of the smaller
set, 0 in this case is not a number. It is the symbol for the
unknown or absence. Hence the result is :
(37 . 4 . 21) / (5 . 4 . 2) paṇas

184 BRAHMAGUPTA AND ARITHMETIC
The above method of working Rule of Three
is found among Arabs, although it does not seem to
have been used in India after Brahmagupta.
Problem Containing Quadratic Equation
Perhaps Āryabhaṭa I is the first man in the history of
mathematics to give a solution of a quadratic equation (499
A.D.). In his Āryabhaṭīya, he gives a rule for the solution of the
following problem (I am reproducing it as described by Datta
and Singh) :
The principal sum p (=100) is lent for one month
(interest unknown = x). This unknown interest is then
lent out for t(=six) months. After this period, the
original interest (x) plus the interest on this interest
amounts to A(=16). The rate-interest (x) on the
principal (p) is required.
This problem requires the solution of the quadratic
equation :—
tx² + px - AP = 0
which gives x = (-p/2 ± √((p/2)² + Apt)) / t
The negative value of the radical does not give a solution
of the problem; so that the result is
x = (√(Apt + (p/2)²) - p/2) / t
This solution is stated by Āryabhaṭa I in the following
words :
Multiply the sum of the interest on the principal and
the interest (A) by the time (t) and by the principal
(p). Add to this result the square of half the principal
(p/2)². Take square-root this. Subtract half the
principal (p/2) and divide the remainder by the time
(t). The result will be the (unknown) interest (x) on
the principal.¹
Here the Sanskrit terms are mūla for principal and phala
for interest.

1. मूलफलं सफलं कालमूलगुणमर्धमूलकृतियुक्तम् ।
   मूलं मूलार्धोनं कालहृतं स्यात्स्वमूलफलम् ॥ Ārya. II. 25,