पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

2 DEFINITIONS
(23) trairâśika (rule of three),
(24) vyasta-trairâśika (inverse rule of three),
(25) pañca-râśika (rule of five),
(26) sapta-râśika (rule of seven),
(27) nava-râśika (rule of nine),
(28) bhâṇḍa-prati-bhâṇḍa (barter of commodities),
(29) jîva-vikraya (sale of living beings) ;
and the nine vyavahâras (determinations) arranged as follows :
(1) miśraka (mixtures),
(2) śreḍhî (series),
(3) kṣetra (plane figures),
(4) khâta (excavations),
(5) citi (piles of bricks),
(6) krâkaca (sawn pieces of timber),
(7) râśi (heaps or mounds of grain),
(8) châyâ (shadow), and then
(9) śûnya-tatva (the mathematics of zero).
DEFINITIONS
Names of notational places :
7-8. Eka, daśa, śata, sahasra, ayuta, lakṣa, prayuta, koṭi,
arbuda, abja, kharva, nikharva, mahâ-saroja, śaṅku, saritpati, antya,
madhya, and parârdha. are each stated to be ten times the pre-
ceding by those who have a knowledge of them.¹
That is to say, the units’ place is called eka, the tens’ place is called
daśa, the hundreds’ place is called śata, and so on.
Lists of names of notational places given by some writers extend
beyond 18 places. Of these, three are as follows :
Mahâvîra’s (850 A.D.) list : (1) eka, (2) daśa, (3) śata, (4) sahasra,
(5) daśa-sahasra, (6) lakṣa, (7) daśa-lakṣa, (8) koṭi, (9) daśa-koṭi, (10) śata-
koṭi, (11) arbuda, (12) nyarbuda, (13) kharva, (14) mahâ-kharva, (15) padma,
¹ Cf. GT, p. 1, vv. 2-3 ; L (ASS), vv. 10-11 ; GK, I, p. 1, vv. 2-3.

DEFINITIONS 3
(16) mahâ-padma, (17) kṣoṇî, (18) mahâ-kṣoṇî, (19) śaṅkha, (20) mahâ-śaṅkha,
(21) kṣiti, (22) mahâ-kṣiti, (23) kṣobha, and (24) mahâ-kṣobha.¹
Yallaya's (1480 A. D.) list : (1) eka, (2) daśa, (3) śata, (4) sahasra, (5)
ayuta, (6) lakṣa, (7) prayuta, (8) koṭi, (9) daśa-koṭi, (10) śata-koṭi, (11) arbuda,
(12) nyarbuda, (13) kharva, (14) mahâ-kharva, (15) padma, (16) mahâ-
padma, (17) śaṅkha, (18) mahâ-śaṅkha, (19) kṣoṇi, (20) mahâ-kṣoṇi, (21)
kṣiti, (22) mahâ-kṣiti, (23) kṣobha, (24) mahâ-kṣobha, (25) parârdha, (26)
sâgara, (27) ananta, (28) cintya, and (29) bhûri.²
Pâvaḷûri Mallikârjuna's list : (1) eka, (2) daśa, (3) śata, (4) sahasra,
(5) daśa-sahasra, (6) lakṣa, (7) daśa-lakṣa, (8) koṭi, (9) daśa-koṭi, (10) śata
koṭi, (11) arbuda, (12) nyarbuda, (13) kharva, (14) mahâ-kharva, (15) padma,
(16) mahâ-padma, (17) śaṅkha, (18) mahâ-śaṅkha, (19) kṣoṇi, (20) mahâ-kṣoṇi,
(21) kṣiti, (22) mahâ-kṣiti, (23) kṣobha, (24) mahâ-kṣobha, (25) nidhi,
(26) mahâ-nidhi, (27) parârdha, (28) parata, (29) ananta, (30) sâgara, (31)
avyaya, (32) aprameya, (33) atula, (34) ameya, (35) bhûri, and (36) mahâ-bhûri.³
Table of money-measures :
9. A purâṇa is equivalent to sixteen paṇas ; a paṇa is
equivalent to four kâkiṇîs ; and a kâkiṇî is equivalent to
twenty varâṭakas (cowries).⁴
Śrîpati's (1039 A.D.) dramma is equivalent to Śrîdhara's purâṇa.⁵
According to Nârâyaṇa (1356 A. D.) :⁶
12 paṇas = 1 dramma,
36 drammas = niṣka.
Table of weights :
10. A mâṣa is equal (in weight) to five guñjâs (Abrus
seeds) ; a weight of sixteen mâṣas is called a karṣa ; a karṣa of
gold is called a suvarṇa ; and four karṣas make a pala.⁷
Nârâyaṇa mentions one more weight, viz., tulâ, which is equal to
100 palas.⁸

¹ See GSS, i. 63-68.
² This list is given in the end of Yallaya's commentary on Â. ii.
³ This list is given in Pâvalûri Mallikârjuna's Gaṇita-śâstra.
⁴ Cf. GT, p. 2, vs. 4 ; L (ASS), vs. 2.
⁵ See GT, p. 2, vs. 4.
⁶ See GK, I, p. 1, vs. 4.
⁷ Cf. GK, I, p. 2, vs. 5.
⁸ See GK, I, p. 2, vs. 5.

4 DEFINITIONS
Table of measures of capacity :
11. A khârî is equal to sixteen droṇas ; a droṇa is equal
to four âḍhakas ; an âḍhaka is equal to four prasthas ; and a
prastha is equal to four kuḍavas.¹
The terms droṇa, âḍhaka and kuḍava are also found to be mentioned
in the Vedâṅga-Jyautiṣa,² where an âḍhaka is defined to be a vessel capable
of holding 50 palas of water.
According to the commentator of the present work there are 3200 palas
in a khârî, so that
1 droṇa = 200 palas,
1 âḍhaka = 50 palas,
1 prastha = 12½ palas,
and 1 kuḍava = 3⅛ palas.
Thus we see that the âḍhaka used in the time and locality of the com-
mentator was the same as that used in the time of the Vedâṅga-Jyautiṣa.
From the Triśatikâ³ of Śrīdhara and the Lîlâvatî⁴ of Bhâskara II
(1150 A D.) it appears that the khârî used in Magadha was equivalent to
a cubic cubit. Nârâyaṇa's khârî is equivalent to 5 cubic cubits.⁵
Table of linear measures :
12. Twenty-four aṅgulas (finger-breadths) make a hasta
(cubit) ; four hastas make a daṇḍa (staff) ; two thousand of
them make a krośa ; and four krośas make a yojana.⁶
Nârâyaṇa's daṇḍa is bigger than Śrīdhara's, being equivalent to 10
cubits ; but his other measures are the same as those given here.⁷
Table of time-measures :
13. Sixty ghaṭîs make a nychthemeron (a day-and-night) ; ⁸
thirty of them make a month ; twelve of them make a year.
These are the definitions (used) in this (pâṭî) gaṇita.
――――――――――――――――――――――――――――――――――――
¹ Cf. GSS, i. 36-37 ; L (ASS), vv. 7-8.
² See Ârca-Jyautiṣa, vs. 17 ; Yâjuṣa-Jyautiṣa, vs. 24.
³ See râśi-vyavahâra.
⁴ See L (ASS), p. 10, vs. 7.
⁵ See GK, I, p 3, vv. 10-13.
⁶ Cf. GSS, i. 29-31(i) ; GK, I, pp. 2-3.
⁷ See GK, I, p. 2, vv. 6 ii)-7(i).
⁸ Thus 1 ghaṭî = 24 minutes.

LOGISTICS 5
LOGISTICS (Parikarma)
(1) Addition (saṅkalita)
Rule for finding the sum of a series of natural numbers :
14(i). When the first term (ādi) and the common differ-
ence (caya) of a series (in arithmetic progression) are (each)
unity, the sum (saṅkalita) is equal to half the number of terms
(pada) multiplied by the number of terms plus one.¹
That is,
Sₙ = 1 + 2 + 3 + ... + n = n(n + 1) / 2 .
Rule for finding the number of terms of a series of natural
numbers when the sum is given :
14(ii). The number of terms (gaccha) is equal to the
(integral) square root of twice the sum of the series, which
must be the same as the residue left (after the extraction
of the square root).
That is,
n = integral part of √(2 Sₙ) .
Ex. 1. What are the sums of 1 to 10, each multiplied
by 10, terms of the series whose first term and common
difference are each unity ? Also, from those sums quickly say
the number of terms of the (various) series.
*‘1 to 10, each multiplied by 10’ means 10, 20, 30, 40, 50, 60, 70, 80,
90 and 100.
Alternative rules :
15. Or, the sum of a series (of natural numbers) is
equal to one-half of—the square of the number of terms plus
the number of terms.
And that (sum) multiplied by 8, (then) increased by 1,
(then) reduced to its square root, (then) diminished by 1,

¹ Cf. GK, I, p. 114, lines 15-16.

6 LOGISTICS
and (then) halved, is the number of terms (in the series).
That is,
Sₙ ≡ 1 + 2 + 3+.... + n = (n² + n) / 2 ,
and n = (√(8 Sₙ + 1) - 1) / 2 .
(2) Subtraction (vyavakalita)
Rule for finding the remainder accruing on subtracting one
series of natural numbers (‘subtrahend series’) from another
series of natural numbers (‘minuend series’) :
16. Having added (nidhāya) the number of terms of the
subtrahend series (vyavakalita-pada) plus one to the number of
terms of the minuend series (saṅkalita-pada), multiply that (sum)
by the difference of the number of terms (of the two series) :
that (product), when halved, becomes the residue of subtrac-
tion (of the given series).
That is,
Sₙ - Sₘ = (n - m) [n+ (m+1)] / 2 ,
where n > m, and Sₖ stands for
1 + 2 + 3 +...+ k .
Ex. 2. What are the respective remainders obtained
when the sums of 1 to 10, each multiplied by 10, terms of the
series whose first term and common difference are (each)
unity are severally subtracted from the sum of 100 terms (of
the same series) ?
Rule for finding the number of terms of the subtrahend series,
when the difference of the minuend and subtrahend series as
well as the number of terms of the minuend series is given :
17. Having subtracted the residue of subtraction (i.e.,
the difference of the minuend and subtrahend series) from
the sum of the minuend series, and multiplied the remainder
(obtained) by 2, the square root thereof, which must be equal

MULTIPLICATION 7
to the residue left (after the extraction of the square root),
should be declared as the number of terms (of the subtrahend
series).
That is, m = integral part of √2[Sₙ - D] ,
where D ≡ Sₙ - Sₘ .
(3) Multiplication (pratyutpanna)
Four methods of multiplication :
18-19. Having placed the multiplicand (guṇya) below
the multiplier (guṇa-rāśi) as in the junction of two doors,
multiply successively in the inverse or direct order, moving
(the multiplier) each time. This process is known as kavāṭa-
sandhi ("the door-junction method").
When the multiplication ( pratyutpanna ) is performed by
keeping that (i.e., the multiplier) stationary, the process is
called tatstha (i.e., "multiplication at the same place") on that
account.¹
20. The process of multiplication called khaṇḍa (or
khaṇḍa-guṇana, "multiplication by parts") is of two varieties
(called rūpa-vibhāga and sthāna-vibhāga), depending on whether
the multiplicand or multiplier is broken up into two or more
parts whose sum or product is equal to it, or the digits stand-
ing in the different notational places (sthāna) of the multipli-
cand or multiplier are taken separately.
These are the four methods of multiplication.²
For details of these methods, see B. Datta and A. N. Singh, History
of Hindu Mathematics, Part I, pp. 136-143, and 146-149.
Ex. 3. Multiply 1296 by 21, 896 by 37, and 8065 by 60.
Operations with cipher :
21. When something is added to cipher, the sum is

¹ See Siṁhatilaka Sūri's comm. on Śrīpati's Gaṇita tilaka, Rule 15-
16(i). The tatstha method of multiplication has been called tatsthāna-guṇana
by Gaṇeśa See L (ASS), p. 17.
² Cf. GSS, ii. 1 ; GT, pp. 4-5, vv. 15-16(i). Also see MSi, xv. 3 ;
SiŚe, xiii. 2 ; GK, I, p. 4, vv. 13-14.

8 LOGISTICS
equal to the additive (kṣepa) ; when cipher is added to or
subtracted from a number, the number remains unchanged.
In multiplication and other operations on cipher, the result is
cipher. Multiplication (of a number) by cipher also yields
cipher.¹
That is, 0 + a = a,
a ± 0 = a,
0 × a or 0 ÷ a = 0,
a × 0 = 0.
(4) Division (bhāgahāra)
Rule for division :
22. Having removed the common factor, if any, from
the divisor and the dividend, divide (by the divisor the digits
of the dividend) one after another in the inverse order : this
is the method of division. ²
Âryabhaṭa II (c. 950 A.D.) gives more details of the process :
“Having placed the divisor underneath the (last digits) of the
dividend, subtract the proper multiple of the divisor from the (last digits
of the) dividend ; the multiple (thus obtained) is the (partial) quotient
Next having moved the divisor (one place to the right) divide what
remains. (Continue this process until all the digits of the dividend have
been taken).”³
(5) Squaring (varga)
Rule for squaring :
23. To obtain the square of a number (proceed suc-
cessively as follows) : Having squared the last digit (antya-
pada) (i.e., having written the square of the last digit over the
last digit), multiply the remaining digits by twice the last,
moving it from place to place (towards the right, and set down
the resulting products over the respective digits); then (rub
out the last digit and) move the remaining digits (one place
to the right).⁴
¹ Cf. GSS, i. 49 ; MSi, xv. 10(ii)-11(i) ; GK, I, p. 13, vs. 30.
² Cf. GSS, ii. 19 ; GT, p. 6, lines 21-24 ; L (ASS), I, vs. 18(ii) ;
GK, I, p. 5, vs. 16(ii).
³ MSi, xv. 4. Also see SiŚe, xiii. 3 ; L (ASS), I, vs. 18(i) ; GK, I,
p. 5, vs. 16(i).
⁴ Cf. GSS, ii. 31 ; GT, p. 7, lines 24-25 ; L (ASS) I, 19 ; GK, p. 6,
vs. 17.

SQUARE-ROOT 9
This rule is based on the formula
(a+b)² = a² + 2ab + b².
For details of the process, see B. Datta and A. N. Singh, History of
Hindu Mathematics, Part I, pp. 157-160
Other rules for squaring :
24. The square (of a given number) is also equal to
the product of two equal numbers (each equal to the given
number), or the sum of as many terms of the series whose first
term is 1 and common difference 2, or the product of the
difference and the sum of the given number and an assumed
number plus the square of the assumed number.¹
That is, (i) n² = n × n ;
(ii) n² = 1 + 3 + 5 +......to n terms ;
(iii) n² = (n - a) (n + a) + a².
Ex. 4. Tell (me) the squares of 1 to 9, 25, 36, 63, 432,
and 7802.
(6) Square root (varga mūla)
Rule for finding the square root :
25-26. Having subtracted the (greatest possible) square
from the (last) odd place (set down double the square root
underneath the next place). By that double square root, that
has left its place (i.e., which has been set down underneath the
next place), divide the remainder ; set down the quotient in
the line (of double the square root), and, having subtracted
the square of that (from the number above), make double of
that (quotient). Then having moved the resulting quantity
(in the line of double the square root) one place forward,
divide by it as before. (Continue this process till all places
are exhausted, and then) halve the doubled quantity (to get
the square root).²
This rule will be clear by the following example :

¹ Cf. GSS, ii. 29 ; GK, I, p. 6, vs. 18. Also see Â. ii. 3(i) ; BrSpSi,
xii 63(ii); MSi, xv. 6(i); GT, p. 8, vs. 21(ii); SiŚe, xiii 4(i); L (ASS), 20(ii).
² Cf. GSS, ii. 36 ; GT, p. 9, vs. 23 ; MSi, xv. 6(ii)-7 ; SiŚe, xiii. 5 ;
L (ASS), p. 21, vs. 22 ; GK, I, p. 7, lines 2-9. Also see Â, ii. 4.

10 LOGISTICS
Example. Find the square root of 186624.
Indicating the odd and even places by writing the tachygraphic
abbreviations o and e over the digits, we have
e o e o e o
1 8 6 6 2 4
Subtracting the greatest square number (i.e., 16) from the last
odd place (i.e. from 18), and then setting down double the square root
of 16 underneath the next place, we have
o e o e o
2 6 6 2 4 (remainder)
8 (line of double the square root)
Dividing out 26 by 8 and setting the quotient 3 in the line of double
the square root, we get
e o e o
2 6 2 4 (remainder)
8 3 (line of double the square root)
Subtracting the square of the quotient (i.e., 9) from above, and
doubling the quotient (3), we get
e o e o
1 7 2 4 (remainder)
8 6 (line of double the square root)
One round of the operation is now over. Now we move 86 one place
forward. Then dividing out 172 by 86 and writing the quotient in the
line of double the square root, we get
o e o
4 (remainder)
8 6 2 (line of double the square root)
Finally subtracting the square of the quotient (i.e., 4) from above,
and doubling the quotient 2, we get
o e o
0 (remainder)
8 6 4 (line of double the square root)
The process now ends. So we divide 864 by 2, getting 432 as the
required square root.
As no remainder is left, the square root is exact.

CUBE 11
(7) Cube (ghana)
Rules for cubing :
27-28. [Let the last digit of the given number be called
the ‘last’ (antya) and the last-but-one digit the ‘first’ (âdi) or
the ‘preceding’ (pûrva).]
Set down the cube of the ‘last’; then set down,
(successively) one place forward (sthânâdhikyaṁ), (i) the square
of the ‘last’ as multiplied by thrice the ‘preceding’, (ii) the
square of the ‘first’ as multiplied by the ‘last’ as well as by 3,
and (iii) the cube of the ‘first’. This gives the cube of the
combined number (formed by the ‘last’ and the ‘first’) (niryukta-
râśi), which should now be treated as the ‘last’ (provided there
be more than two digits in the given number).¹
The (continued) product of three equal quantities ;² or,
in the series having 1 for the first term and common difference
(considering the last two terms, designating them as the ‘first’
or ‘preceding’ and the ‘last’ respectively), the last multiplied
by thrice the ‘first’, and increased by 1, and that added to the
cube of the ‘preceding’, is also the cube.³
Of the three rules stated here, the first one is the main method of
cubing a number. To illustrate it by an example, let us find the
cube of 256.
To begin with we treat 2 as the ‘last’ and 5 as the ‘first’. Then
writing the cube of the ‘last’ (i.e., 2³, or 8), then in the next place the
square of the ‘last’ as multiplied by thrice the ‘first’ (i.e., 3×5×2², or 60),
then in the next place the square of the ‘first’ as multiplied by thrice the
‘last’ (i.e., 3×2×5², or 150), and then in the next place the cube of the
‘first’ (i.e., 5³, or 125), we have
0 0 0 0
8
6 0
1 5 0
1 2 5

¹ Cf. BrSpSi, xii. 6 ; GSS, ii. 47 ; GT, p. 11, vs. 25 ; L (ASS), p. 23,
vs. 24-25(i) ; GK, I, p. 7, lines 17-19, p. 8, lines 1-2.
² Cf. Ā, ii. 3(ii) ; BrSpSi, xii. 62(ii) ; GSS, ii. 43(i) ; MSi, xv. 6(i) ;
GT, p. 11, line 10 ; SiŚe, xiii. 4(i) ; L (ASS), p. 23, vs. 24(i) ; GK, I, p. 7,
line 16.
³ Cf. GT, p. 11, lines 7-9 ; GK, I, p. 8, lines 3-4.

12 LOGISTICS
Adding up these numbers, we get 15625. This is the cube of 25.
Now we treat 25 as the last and 6 as the first, and proceeding as
above we get 16777216 as the cube of 256.
The second and third rules are :
(i) 𝑛³ = 𝑛 × 𝑛 × 𝑛
(ii) 𝑛³ = (𝑛 - 1)³ + 3𝑛 (𝑛 - 1) + 1 .
Datta and Singh, assuming
खैकादिचये वाऽन्त्ये त्र्यादिहते पूर्वघनयुतिः सैके ॥
to be the reading of the second half of verse 28, have given the following
translation for it : ¹
"The cube (of a given number) is equal to the series whose terms
are formed by applying the rule, 'the last term multiplied by thrice the
preceding term plus one,' to the terms of the series whose first term is zero,
'the common difference is one and the last term is the given number."
This gives
𝑛³ = Σ [ 3𝑟(𝑟 - 1) + 1 ]
in place of formula (ii) above.²
Ex. 5. Quickly say what are the cubes of 1 to 9, 15,
256, and 203.
(8) Cube root (ghana-mûla)
Rule for finding the cube root :
29-31. (Divide the digits beginning with the units' place
into periods of ) one 'cube' place (ghana-pada) and two 'non-
cube' places (aghana-pada). Then subtracting the (greatest pos-
sible) cube from the (last) 'cube' place and placing the (cube)
root underneath the third place (to the right of the last 'cube'
place), 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' (âdima) and the cube
root as the 'last' (antya)], subtract the square of that quotient,
as multiplied by thrice the 'last' (antya), from one place less
than that occupied by the quotient (uparima-râśi) as before, and

¹ B. Datta and A. N. Singh, History of Hindu Mathematics, Part I,
page 168.
² This formula is also given in GSS, ii. 45.

CUBE ROOT 13
the cube of the ‘first’ (âdima) 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 cube place from the left).
Again apply the rule, ‘(placing the cube root) under the third
place’ etc. (provided there be more than two ‘cube’ places in
the given number ; and continue the process till all cube
places are exhausted). This will give the (cube) root (of the
given number).¹
To illustrate this method, we find out the cube root of
277167808.
Indicating the ‘cube’ and ‘non-cube’ places by writing the tachy-
graphic abbreviations c and n over the digits, we have
n n c n n c n n c
2 7 7 1 6 7 8 0 8
Subtracting the greatest possible cube (i.e., 6³ or 216) from the last
‘cube’ place (i e., from 277) and placing the cube root (i.e., 6) underneath
the third place to the right of the last ‘cube’ place, 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 the quotient 5 be called the ‘first’ and the cube root 6 the
‘last’. Then subtracting the square of the ‘first’ as multiplied by thrice
the ‘last’ (i.e., 3×6×5², or 450) from one place less than that occupied by
the quotient (i.e., from 716), we get
n n c n n c n n c
2 6 6 7 8 0 8 (remainder)
6 5 (line of cube root)

¹ Cf. Â, ii. 5 ; BrSpSi, xii. 7 ; GSS, ii. 53, 54 ; MSi, xv. 9-10(i) ;
GT, p. 13, lines 18·25 ; Siśe, xiii. 6-7 ; L (ASS), vv. 28-29 ; GK, I, pp. 8-9,
vv. 24-25.

14 LOGISTICS
And subtracting the cube of the 'first' (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 number 65 stand-
ing in the line of the cube root is the cube root of the given number
(277167808) upto its last-but-one 'cube' place from the left (i.e., of 277167).
As there is one more 'cube' place 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, 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) from 780 we get
n n c n n c n n c
8 (remainder)
6 5 2 (line of cube root)
Finally subtracting 2³ 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 the operation is now over. There being no
more 'cube' places on the right, the process ends. The quantity in the line
of the cube root, viz., 652, is the cube root of the given number. The
remainder being zero, the cube root is exact.

OPERATIONS FOR FRACTIONS 15
(9-16) Operations for fractions
Rule for the addition of fractions :
32(i). Reduce the fractions to a common denominator
and then add the numerators. The denominator of a whole
number is unity.¹
Ex. 6. Say the sum of ½, ⅓, ⅙ and ¹/₁₂, and of 2 plus ½,
3 minus ¼, and 6.
Ex. 7. Friend, if you know the method of calculation
(gaṇita-vidhi), quickly say the sum of 1½ terms, ½ term, and of
⅓ term of the series whose first term (ādi) and common
difference (caya) are each unity.
Rule for the subtraction of fractions :
32(ii). Take the difference of the numerators of the
positive and negative fractions reduced to a common denomi-
nator.²
Ex. 8. Subtract ¼, ⅓ and ⅙ from 1 and say what remains.
Also, subtract 3 — ½ and 2+⅓ from 5 (and say the remainder).
Ex. 9. Say what remains as the remainder when the
sum of 2 plus ½ terms is subtracted from the sum of 5 plus ½
terms of the series whose first term and common difference are
(each) unity.
It may be remarked here that, according to the author of the
present work, rule 14(i) applies also to series of natural numbers
having a fractional number of terms. Series in A. P. having frac-
tional number of terms have generally no meaning but they may be
interpreted geometrically by means of figures called series-figures. See
infra, ‘Determinations pertaining to series’ (śreḍhī-vyavahāra).
¹ Cf. MSi, xv. 14(i) ; GT, p. 15, lines 20-21 ; SiŚe, xiii. 8 ; L
(ASS), p. 35, vs. 37 ; GK, I, p. 11, lines 6-7. Also see BrSpSi, xii. 2.
² Cf. BrSpSi, xii. 2(ii); MSi, xv. 14(ii) ; GT, p. 18, lines 3-4 ;
SiŚe, xiii. 8 ; L (ASS), p. 35, vs. 37; GK, I, p. 11, lines 6-7.

16 LOGISTICS
Rule for the multiplication of fractions :
33(i) The product (of the given fractions) is obtained on
dividing the product of the numerators by the product of the
denominators.¹
Ex. 10. 2 plus ½ is multiplied by 1 plus ½, and 60 plus ⅓
is multiplied by ⁵⁄₂ : what are the products, say separately.
Rule for the division of fractions :
33(ii). Having interchanged the denominator and the
numerator of the divisor, apply the previous rule (i.e., the
rule for multiplication).
Ex. 11. 6 plus ¼ is divided by 2+½, and 60 plus ¼ is
divided by 3+⅓ : say the quotients separately.
Rule for squaring a fraction :
34(i). The square of the numerator divided by the
square of the denominator is the square of the fraction.³
Ex. 12. Say, friend, if you know, the square of 2 plus ½,
of 15 plus ¼, of ½, and of ⅓.
Rule for finding the square root of a fraction :
34(ii). The square root of the numerator divided by
the square root of the denominator, gives the square root
(of the fraction).⁴
Rule for cubing a fraction :
35(i). The cube of the numerator divided by the cube
of the denominator gives the cube (of the fraction).⁵
¹ Cf. BrSpSi, xii. 3(ii) ; GSS, iii. 2; MSi, xv. 15(i) ; GT, p. 19,
line 26 ; SiŚe, xiii. 9(i) ; L (ASS), p. 36, vs. 39 ; GK, I, p. 12, lines 2-3.
² Cf BrSpSi, xii. 4 ; GSS, iii. 8(i) ; MSi, xv. 15(ii) ; GT, p. 21,
line 3-6 ; SiŚe, xiii. 10 ; L (ASS), p. 37, vs. 41 ; GK, I, p. 12, lines 11-12.
³ Cf. BrSpSi, xii. 5(i) ; GSS, iii. 13 ; MSi, xv. 16(i) ; GT p. 22,
line 24; SiŚe, xiii. 9(ii); L (ASS), p. 38, lines 2-3; GK, I, p. 12, lines 17-18.
⁴ Cf. BrSpSi, xii. 5(ii) ; GSS, iii. 13 ; MSi, xv 16(ii) ; GT, p. 23,
line 25 ; L (ASS), p. 38, lines 2-3 ; GK, I p. 12, lines 17-18.
⁵ Cf. GSS, iii. 13 ; MSi, xv. 17(i) ; GT, p. 25, line 13 ; L (ASS), I,
p. 38, lines 2-3 ; GK, I, p. 12, lines 17-18.

REDUCTION OF FRACTIONS 17
Ex. 13. Say, if you know, the cube of 7 plus ½ ,
of 17 plus ¼ , of ¼ , and of ⅕ .
Rule for extracting the cube root of a fraction :
35(ii). The cube root of the numerator divided by the
cube root of the denominator gives the cube root (of the
fraction).¹
(17-22) Reduction of fractions (kalâ-savarṇa)
Rule for reducing fractions of the bhâga class (i.e., fractions
connected by + and – signs):
36. In the bhâga class, in order to reduce the (two given)
fractions to a common denominator, remove the common
factor, if any, from the denominators, and then by each of
them (i.e., the reduced denominators) multiply the denomina-
tor and numerator of the other fraction.²
Ex. 14. What sum is obtained by adding together the
fractions having (the integers) 2 to 6 for their denominators,
and 1 for their numerators, and by adding together the frac-
tions having (the integers) 3 to 9 for their denominators and
(the integers) 2 etc. for their respective numerators.
Alternative rule for reducing fractions of the bhâga class :
37. By the lower denominator multiply the upper
numerator, (then) by the upper denominator multiply the
lower denominator, and (then) add the product of the
numerator and the denominator in the middle to the upper
numerator.³
Suppose, for example, that we want to reduce the fraction
2/3 + 4/5
───────────────────────────────────────────────────────────
¹ Cf. GSS, iii. 13 ; MSi, xv. 17(ii) ; GT, p. 26, line 27 ; L (ASS),
p. 38, lines 2-3 ; GK, I, p. 12, lines 17-18.
² Cf. BrSpSi, xii. 2(i) ; MSi, xv. 13(ii) ; GT, p. 30, line 16 ;
SiŚe. xiii. 11(i) ; L (ASS), p. 28, line 9 ; GK, I, p. 9, vs. 26(i). Also see
GSS, iii. 55 (i);
³ Cf. SiŚe, xiii. 12.

18 LOGISTICS
Writing these fractions one below the other without the lines of
separation as the Hindus used to do, we get
2
3
4
5
[In this scheme, 2 is the upper numerator and 3 the upper denomi-
nator ; similarly 4 is the lower numerator and 5 the lower denominator.]
Now according to the rule, multiplying the upper numerator by the
lower denominator, and multiplying the lower denominator by the upper
denominator, we get
2 × 5      10
  3        3
     i.e.,
  4        4
5 × 3      15
Next, adding the product of the numerator and the denominator in
the middle (viz. 3 and 4) to the upper numerator, we get
(10 + 12) / 15   i.e.,   22 / 15   or   22 / 15
rubbing out the figures in the middle, which are not required.
Rule for reducing fractions of the prabhāga class (i.e., fractions
connected by 'of') :
38(i). In the prabhāga class, one should multiply the
numerators together, and the denominators together.¹
That is,
(a / b) of (c / d) of (e / f) = ace / bdf .
Ex. 15. Tell (me) the sum of
1/4 of 1/2 of 1/2 , 1/10 of 1/6 of 1/5 of 1/3 , and 1/7 of 1/6 of ( 2 + 1/2 ).
Rule for reducing a fraction of the bhāga-bhāga class (i.e., a
whole number divided by a fraction) :
38(ii). Having multiplied the whole number (rūpa) by
the denominator of the fraction, remove the denominator :
¹ Cf. BrSpSi, xii. 8(ii) ; GSS, iii. 99 ; MSi, xv. 13(i) ; SiŚe, xiii.
11(i) ; L (ASS), p. 31, vs. 32 ; GK, I, p. 9, vs. 26(ii).
Indological Truths

REDUCTION OF FRACTIONS 19
this is the process (of reduction) for the bhāga-bhāga class.¹
That is,
a ÷ b/c = ac / b .
Ex. 16. Friend, if you know (the method), say after
thinking, what sum will be obtained by adding together (the
fractions) :
1 ÷ 1/6 , 1 ÷ 1/10 , 1 ÷ 1/3 , 1 ÷ 1/9 , and 1 ÷ 1/2 .
Ex. 17. 1 has been severally divided by fractions
having (the integers) 3 to 6 for their denominators and (the
integers) 2 etc., for their respective numerators. Say what
sum will be obtained when they are added together.
Rule for reducing a fraction of the bhāgānubandha class (i.e., a
whole number increased by a fraction, or a fraction increa-
sed by a fraction of itself) :
39. In the bhāgānubandha class, the whole number
(rūpagaṇa) as multiplied by the denominator (of the fraction)
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 multiplied by the sum of the lower numerator and
denominator.²
That is,
(i) a + b/c = (ac + b) / c ,
(ii) a/b + c/d of a/b ( which was written in the Hindu
way as³ [ (a/b) / (c/d) ] ) = a (d + c) / bd .
─────────────────────────────────────────────────────────────
¹ Cf. BrSpSi, xii. 9(i) ; GSS, iii. 99(ii) ; MSi, xv. 19(i).
² Cf. BrSpSi, xii 9(ii) ; GSS, iii. 113 ; MSi, xv. 11(ii)-12 ; GT,
p. 34, lines 15-16 ; L (ASS), p. 32, vs. 34 ; GK, I, p. 9, vs. 27.
³ Actually it was written as a
b
c
d

20 LOGISTICS
Ex. 18. What sum is obtained by adding together
1+½, 5+¼, and 8+⅓ ?
Ex. 19. What is the sum of
(3 + ½) + ¼ of (3 + ½) + ⅙ of {(3 + ½) + ¼ of (3 + ½)}
and ½ + ⅓ of ½ + ¼ of {½ + ⅓ of ½} ?
Rule for reducing a fraction of the bhāgāpavāha class (i.e., a
whole number minus a fraction, or a fraction minus a
fraction thereof) :
40. In the bhāgāpavāha class, the numerator of the frac-
tion should be subtracted from the whole number multiplied
by the denominator (of the fraction) ; or having multiplied
the upper denominator by the lower denominator, the upper
numerator should be multiplied by the lower denominator as
diminished by the lower numerator.¹
That is to say,
(i) a - b / c = (ac - b) / c ,
(ii) a / b - c / d of a / b (which was written in the Hindu way
a
b
as² ) = a (d - c) / bd .

• c
  d
  Ex. 20. Say the amount, when 1 - ½ , 5 - ¼ , and 8 - ⅓
  are added together.
  Ex. 21. What is obtained by adding together :
  (3 - ½) - ¼ of (3 - ½) - ⅙ {(3 - ½) - ¼ of (3 - ½)}
  and ½ - ⅓ of ½ - ¼ of (½ - ⅓ of ½) ?
  ──────────────────────────────────────────────────────────
  ¹ Cf. BrSpSi, xii. 9(ii) ; GSS, iii. 126 ; MSi, xv. 11(ii)-12; GT, p. 37,
  lines 2-5 ; L (ASS), p. 32, vs. 34 ; GK, I, p. 9, vs. 27.
  ² Actually, it was writtten as
  a a
  b or b
  c+ ˙c
  d d
  the fractions being written without the separating line, and + or (˙) being
  written to denote subtraction.

REDUCTION OF FRACTIONS 21
The commentator interprets ‘rûpatrayamardhonam’ as meaning
3 - ½ of 3 and not (3 - ½), so the first expression according to him is
(3 - ½ of 3) - ½ of (3 - ½ of 3) - ⅓ of [(3 - ½ of 3) - ½ of (3 - ½ of 3)].
The interpretation given by us is more natural and plausible,
especially when we see that a similar interpretation has been given to
the verse of Ex. 19, which is analogous to this one.
Rule for reducing a chain (vallî) of measures :
41. To reduce a chain (vallî), multiply the foremost
denominator and numerator by the lower denominator, and
(then) from or to the foremost numerator subtract or add
(as the case may be) the lower numerator.¹
Ex. 22. What amount is obtained by reducing
5 purâṇas , 3 paṇas , 1 kâkiṇî , - 1 varâṭaka , - ⅕ of that (i.e., of
a varâṭaka) (to purâṇas) ?
Converting each measure to the previous one, and writing them one
below the other in the Hindu way, we have
5
1
3
16
1
4
1+
20
1+
5
where + indicates subtraction.
Now applying the rule to the uppermost two fractions of this chain,
we get
83
16
1
4
1+
20
1+
5
¹ Cf. MSi, xv. 18 ; GT, p. 39, lines 7-10.