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

ग्रन्थेऽस्मिन् प्रयुक्तपारिभाषिकशब्दानाम्
अनुक्रमणिका
अंश २३, २६, २८-३२, ३४, अवनि १०६
३५, ७१ अवलम्ब १०६, १५६
अंश (=सुवर्णखण्ड) ७१ असदृशलम्ब १७५
अंश (=भाग) ७५ असमलम्ब १७५
अंशक २६ असमानचतुर्बाहु १७०
अग (=७) ७२ अहोरात्र ६
अगुण ८० आढक ५
अघनपद २१ आदि १६, ११०, ११३, ११८, १२०,
अङ्गुल ६, ४०, १६४, १७० १२५, १२६, १३०, १३४, १३६,
अनष्ट १२६, १३० १३६, १४१, १४५, १४८, १५३
अनुलोममार्ग १३ आद्य ४२
अन्तर्मृति ६६ आयतसमचतुरश्र १५६, १६१
अन्त्य १६, २१, ४२, १३७ आयराशि २५
अन्त्य (अंकस्थानसंज्ञा) ५ आयाम ४४, ४८, ४६
अन्त्यपद १६ आवर्त ६८
अन्त्यार्घ ७६ आहुति १३६
अन्यजाति ३७ इच्छा ३७, ८०
अन्यपक्ष ४५ इष्ट १६, १०६, १४१
अपगम १४ इष्टादिचय १०७
अब्ज ५ इष्टावलम्ब १०६
अभिन्नजाति ३७ ईप्सितकाल ५५
अभ्यस्त २६ उत्तर ११३, १२०, १३६, १४१, १५३
अभ्यास २०, ३१, ४५, १०५ उत्सार्य १३
अयुत ५ उद्धृत ५८, ७१
अर्ध ७७-७६ उपनय ५५
अर्बुद ५ ऋजुगति १५६
अवधा १५६ ऋण ३५

, १४८, १५२,
Right: गुणराशि १३
Left: १५६, १६७
Right: गुणित ४२, ७१
Left: कृतिसङ्कलित १५२
Right: गुण्य १३

Everything aligns perfectly line by line.
Looks great!१७८ अनुक्रमणिका
एक ५ गजदन्त १५६, १७१
एकदिनगति ४१ गजदन्ताकृति १७१
एकपत्र ५६, ६० गणित १, ६, ४२, ११०, १२६, १३०,
एकाद्युत्तरविधि ६७ १३६
कनक ६८ गणित (=क्षेत्रफल) १६१, १६५, १७०,
कर ६, १६१, १६४, १६५, १७०-१७३ १७१, १७२, १७५
कर्ष ५, ३७ गतकालफल ५८
कलासवर्ण २ गति ४१, ८४, ८६, ८७, १३६
कल्याणसुवर्ण ४७ गति (=सङ्कलित) १३६
कवाटसन्धि १३ गतिविशेष ८७
काकिणी ५, ३५ गम (=घटाना) १०२
काञ्चन ६७ गुञ्ज ५
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अनुक्रमणिका १७६
चाप १५६ निखर्व ५
चिति २ निजकाल ५३
छायां २ निरंशरूप १०३
छेद २६, २८, २६, ३१, ३२, ३४, ३५, नियुक्तराशि १६
४५, ७८ निर्विकल्पपद १२६, १२८
छेदगम ३१, ७६ नेमि १५६
छेदन २३ नेम्याकृति १७२
छेदमूल २८ नेम्याकार १७१
छेदसमत्व ७८
पक्व ६४
जीवविक्रय २ पङ्क्ति (=१०) ७२
पञ्चसप्तनवराशि २
तत्स्थ ३१ पण ५, ३५, ३७, ३८, ४८, ५०, ७५,
ताडित ६८ ८८, ११६, १३७
तुल्यच्छेद २५, २८, ७८ पष्य ७५, ७६
तुल्यत्रिभुज १६४ पद (=मूल) १०३, १०५, १०६, १७५
तुल्यहर ७६ पद (=गच्छ) ६, ७, ६, ११, १०८, ११०,
त्रिचतुर्भुज १५६ ११३, ११८, ११६, १२५, १२८,
त्रिभुज १७१, १७२ १२६, १३४, १४१, १४५, १४८,
त्रिराशि ३७ १५०, १५३
त्रैराशिक २ पद (=स्थान) १६, २१
पदवर्ग १५०, १५१
दण्ड ६ परार्द्ध ५
दल ३७ परिकर्म २
दश ५ परिपक्ववह्निभव ६६
दिवाकर (=१२) ७२
दृश्य १०१, १०३, १०५ पल ५, ३७, ३८, ६७, ७६, ८८
दैर्घ्य ४६ पाद ३३, ३४
द्रोण ३६, ४८, ४६ पार्श्वभुज १५६, १६१, १७०
द्वित्रिसमभुज १५६ पिण्ड ४६
द्विसमत्र्यश्र १६५ पुराण ५, ३५, ५२
पूर्वाप्र १०५
घन ३५, १२८-१३० प्रक्षेप ७३
घनयोग ५८
घरा १०८, १०६, १६२, १६७, १७१ प्रचय ११६, १२०, १४५, १५३

१८० अनुक्रमणिका
प्रतिबाहु १५६ भागापवाहजाति ३४
प्रतिलोम १४ भागोनरूप १०४
प्रत्युत्पन्न २, १३, २६ भाटक ८७, ८८, ९५, ११६
प्रत्युत्पन्नविधान १३ भाण्ड ६५
प्रभव १२८, १४५ भाण्डप्रतिभाण्ड ५०
प्रभाग २ भाव्यक ५५
प्रभागजाति ३१ भास्कर (= १२) ७२
प्रमाण ३७, ५४, ६० भिन्न २
प्रमाणराशि ५३ भुज १५६, १६२, १७५
प्रयुत ५ भू १०८–११०, १५६, १६१, १७०,
प्रस्थ ५, ७३, ७४ १७२
फल ३७, ४५, ४६, ५३–५५, ५८, ६०, भूमि १०८, १६१, १६४, १६५
७३, ७४, ८८, १०७, ११०, ११६,
१३४, १३६ मध्य ४२
फल (=सङ्कलित) १२०, १२५, १४१, मध्य (अङ्कस्थानसंज्ञा) ५
१४५, १५३ मध्यमलम्ब १६१, १७०
फल (=क्षेत्रफल) १६१, १६२, १६५, १६७, मध्यलम्ब १६१
१७३ मध्यविस्तर १७२
बालेन्दु १७१, १७२ मध्यव्यास १७३
बाहु १०८, १५६, १६१, १६५, १६७ मध्यायाम १७२, १७३
ब्रीहि ४८ महासरोज ५
मानान्तर ४२
भक्त २६, २७, ४२, ६०, ७१ मूल २, ८, १०, १२, १८, २१, २७
भाग २, १४, ३६, ७१, ९४ २८, ६५, १०३–१०५, १२०,
भागजाति २८ १३०, १७५
भागभाग २, ३१ मूल (=मूलधन) ४६, ५३–५५,
भागभागविधि ३१ मूलासन्न १०३
भागमाता २, ३६ मूल्य ५०, ७५
भागहार २
भागहारविधि १४ यव ६६
भागानुबन्ध २ युत ६२
भागानुबन्धजाति ३२ युति १६, २३, ७१, ७३, १७५
भागापवाह २ युतिवर्ण ६८
योग ३३, ३४, ६३, ८६, ८७

अनुक्रमणिका २८
योजन ६, ४०, ४१, ४८, ८५, ८६, वर्ष ६, ४६
८८, ११६, १३६ वल्याः सवर्णन ३५
रक्तिक ४३ बहुराशिपक्ष ४५
राशि १४ विकल १२६, १२८, १३०
राशि (व्यवहार) २ विपक्व ६६
रूप १० विपक्वकनक ६४
रूपविभाग १३ विपरीत १०६
रूपादिचय ६७ विपरीतीकृत ५०
लक्ष ५ विभाज्य १४
लम्ब १०८-११०, १५६, १६१, १६२, विलोमगति १३
१६५, १६७, १७१, १७२ विवर ११, १४१
लम्बक १०७, १०९, १६४ विश्लेष १४१
लेखक ५५, विषम १३४
वक्त्र १०८, १६१ विषमचतुरश्र १५६
वक्रगति १५६ विषमपद १८
वज्र १७१, १७३ विषमबाहु १७५
वज्रवत् १०८ विष्कम्भ ४४
वदन १०६, १५३, १६१, १६५, १६७, विस्तार १०७
१७३ वृत्त १५६
वध १६, ३०, ३४, ६४, ६७, ६८ वृत्ति ५५
वराटक ५, ३५ वृद्धि (=चय) १४५
वर्ग २, ६, १६, १८, २१, २७, ६५, वृद्धि १२६, १३६, १४८, १५२, १५३
१०३, १०६, १२५ १५०, १५३, वृद्धिधन ५३
१७५ व्यत्यास ४५
वर्गमूल १७५ व्ययराशि २५
वर्गयुति १५२ व्यवकलित २, ११, २५
वर्गसङ्कलित १५०, १५३, व्यवकलितपद ११
वर्ण ६३, ६४, ६६-६९, ७१, ७२ व्यवकलितशेष ११, १२
वर्णक ४३, ४४, ४७, ६३, ६४, ६६, व्यवहार २
६७, ७१ व्यस्त ७१
वर्णमालिका ६६ व्यस्त (त्रैराशिक) २
व्यास ४८, ४६

१८२ अनुक्रमणिका
शङ्कु ५ समलम्ब १७०
शत ५ समलम्बकचतुरश्र १६१
शर (=५) ७२ समहर ७६
शलाका ६६ समायोग २६
शून्यतत्व २ समावर्त ६७
शेष १०१, १७५ समास १६, ३३, ३५, ३६, ५८, १६१
शेषपद १६, २५ समुच्छ्रय ४६
शेषांश १०५ सम्भक्त २६
श्रीधराचार्य १ सरितां पतिः ५
श्रेढी २, ११० सहस्र ५, ६
श्रेढीक्षेत्र १०७, १०९, ११० सहित ३३
श्रेढीगणित १५३ सुवर्ण ५, ४०, ४३, ४४, ४७, ६४, ६७,
श्रेढीफल १५३ ६८
संगमकाल ८७ सूक्ष्मफल १५६
संयोग ३४ सेतिक ४३
संयोगवर्ण ६८ स्थानविभाग १३
सङ्कलित २, ६-९, ११, २४, २५, १२६, स्थूलफल १५६
१४८, १५२, १५३ स्वपादरहित ३४
सङ्कलितकृति १५१ स्वपादसहित ३३
सङ्कलितपद ११, १२ स्वयुति ५३
सङ्कलितसङ्कलित १५१, १५३ हत ६६
सङ्कलितसमास १५२ हति २८
सङ्कलितैक्य १५१ हनन ३०
सङ्गुणन १४ हर १४, २६, २९, ३०, ३२, ३४, ६७,
सङ्गुणित १०, १६१ १०५
सदृशच्छेद २३ हस्त ६, १६१, १६५, १७०-१७३
सदृशद्विराशिघात १६ हीन ३४
सम १३४ हृत ५३, ५८, ६६, ७१, ७२, ७५, १०४
समधन ७६ हेम ६३, ६७-६९, ७१
समचतुरश्र १६१

English Translation
OF THE
PATIGANITA
𝓘𝓷𝓭𝓸𝓵𝓸𝓰𝓲𝓬𝓪𝓵 𝓣𝓻𝓾𝓽𝓱𝓼

SRIDHARA'S
PATIGANITA
Homage and introduction :

1. Having paid obeisance to the Unborn God, the cause
   of creation, preservation and destruction of the worlds, I shall
   briefly state the (pâṭî)gaṇita for the use of the people.
   Contents:
   2-6. (The topics dealt with) here are the twenty-nine
   parikarmas (logisties) arranged as follows :
   (1) saṅkalita (addition),
   (2) vyavakalita (subtraction),
   (3) pratyutpanna (multiplication),
   (4) bhâgahâra (division),
   (5) varga (square),
   (6) varga-mûla (square root),
   (7) ghana (cube),
   (8) ghana-mûla (cube root),
   (9-16) the same (operations) for fractions,
   (17-22) reduction of fractions (kalâ-savarṇa) of six varieties,
   viz.,
   (i) bhâga (fractions connected by + or - ),
   (ii) prabhâga (fractions connected by “ of ”),
   (iii) bhâga-bhâga (a whole number divided by a frac-
   tion),
   (iv) bhâgânubandha (a whole number increased by a
   fraction, or a fraction increased by a fraction
   of itself),
   (v) bhâgâpavâha (a whole number diminished by a
   fraction, or a fraction diminished by a fraction
   of itself), and
   (vi) bhâgamâtâ (a blending of two or more fractions of
   previous forms),

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.