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

xvi INTRODUCTION
cause it found very few applications in everyday life. The
mathematics of zero which Śrīdharācārya listed as the ninth
determination, is Śrīdharācārya’s own contribution. Unfortu-
nately this topic is missing from our manuscript.
In mixtures of things (miśraka), Śrīdharācārya deals with
(i) simple interest, (ii) alligation of gold, (iii) partnership, (iv)
purchase and sale, (v) meeting of two travellers, (vi) wages
and payments, (vii) the well-known cistern problem, (viii)
wages paid out of the commodity, (ix) combination of savours,
and (x) certain special problems reducing to the solution of
simple and quadratic equations. In series (śreḍhī) he deals
with arithmetic and geometric series as well as with series of
squares, cubes, and successive sums of series in arithmetic
progression. In the surviving part of the section dealing with
plane figures, he gives rules for finding the areas of triangles
and quadrilaterals.
The following are the special features of the Pāṭīgaṇita :
(1) Rule for reducing a chain of measures. (Rule 42)
This rule is not found to occur in the Āryabhaṭīya
Brāhma-sphuṭa-siddhānta, Gaṇita-sāra-saṅgraha, or in any
other work written before the tenth century A.D., and
seems to have been first introduced by Śrīdharācārya.
Udayadivākara (1073 A.D.) ascribes it to Śrīdharācārya.
Its occurrence in the Mahā-siddhānta of Āryabhaṭa II
. (c. 950 A.D.) and the Gaṇita-tilaka of Śrīpati (1039 A.D.)
is evidently due to Śrīdharācārya’s influence.
(2) A special rule for finding the time in which a sum lent
out on simple interest will be paid back by equal monthly
instalments. (Rule 49-50). Also the example set on this
rule. (Ex. 55-56).
This rule and example, too, do not occur in any
work written before the tenth century A.D. and seem to
be due to Śrīdharācārya. The occurrence of this rule and

INTRODUCTION xvii
a similar example in the Gaṇita-kaumudî of Nârâyaṇa
proves Śrîdharâcârya’s influence on Nârâyaṇa.
(3) Interpretation of arithmetic series both geometrically and
symbolically. (Rules 79-93)
This is the only work where we find this dual treat-
ment of the subject.
(4) Rule telling us how two travellers starting at different
times with different speeds and accelerations would meet
two times on the way. (Rule 97-98)
This rule again is probably due to Śrîdharâcârya,
and reappears in the Gaṇita-kaumudî of Nârâyaṇa under
the influence of the former.
Besides the above-mentioned specialities which relate to
the subject matter of the Pâṭîgaṇita, there is one more pecu-
liarity which pertains to the composition of the Pâṭîgaṇita and
deserves special notice. This lies in the fact that the
author has taken care to employ exclusively the âryâ metre in
the formation of verses giving the rules. There is not a single
rule which is in any other metre. This, however, cannot be
said regarding the verses containing the examples; although
the examples are also generally set in the same metre. We
find that out of 133 examples, 104 are in the âryâ metre, 26 in
the anuṣṭup metre, 1 in upajâti, 1 in śârdûlavikrîḍit, and 1 in
vasantatilakâ.¹
3·2. The Patiganita-sara ( or Trisatika ). The Pâṭî-
gaṇita-sâra, as its name implies and as the author himself
says in the opening stanza to that work, is an abstract of the

¹ Exs. 1-15, 17-31, 33,.36-44, 47, 50-56, 59-66, 69, 72, 76-77, 86-88,
91-103, 106-133 are in the âryâ metre; Exs. 34, 35, 45, 46, 48, 49, 57, 58
67, 68, 71, 73-75, 78-85, 89, 90, 104, 105 are in anuṣṭup; Ex. 16 is in
upajâti; Ex. 32 is in Śârdûlavikrîḍit; and Ex. 70 is in vasantatilakâ.

xviii INTRODUCTION
Pâtīgaṇita. It gives the important rules and examples of the
Pâtīgaṇita with slight modifications, alterations and additions
here and there, and provides a short course of arithmetic and
mensuration. From the study of the latter part of this work,
which does not have its counterpart in our incomplete manu-
script of the Pâtīgaṇita, we find that the following two approxi-
mate formulæ are Śrîdharâcârya’s own contributions to the
subject :
(1) Area of the segment of a circle
= (√10 / 3) × ((chord + arrow) / 2) × arrow .¹
This formula occurs for the first time in this work and is
better than the following formula given by Mahâvîra² :
(√10 × chord × arrow) / 4
It is noteworthy that Mahâvîra has called this formula
accurate (sûkṣma).
(2) Volume of a sphere = ((diameter)³ / 2) (1 + 1/18)
= 4 (radius)³ (1+1/18)
= (4·22 . . )r³, where r = radius.³
The formulæ given by the earlier writers are as follows :
(i) Âryabhaṭa I’s formula⁴ : (πr²)³ᐟ² = (5·56....)r³.
(ii) Formula⁵ quoted by Bhâskara I : (9/2)r³ = 4·5r³.
(iii) Mahâvîra’s accurate formula⁶ : 9/2 × 9/10 r³ = 4·05r³.
Śrîdharâcârya’s formula is the best, as it more closely
approximates to the modern formula :
(4/3)πr³ = (4·188....)r³.
¹ Triś, p. 35, Rule 47.
² See GSS, vii. 70½.
³ See Triś, p. 39, Rule 56.
⁴ See Â, ii. 7.
⁵ See Bhâskara I’s comm. on Â, ii. 7.
⁶ GSS, viii. 28½.

INTRODUCTION xix
The abovementioned formulæ of Śrîdharâcârya re-ap-
pear in the Mahâ-Siddhânta of Âryabhaṭa II (c. 950 A.D.),
evidently under the influence of Śrîdharâcârya.
3·3. Criticism of Brahmagupta. In none of the
above two works does Śrîdharâcârya refer to any ancient
writer or work by name. At one place in the Pâṭîgaṇita,¹
however, he refers to an ancient rule and finds fault with it.
He writes :
स्थूलफलं त्रिचतुर्भुजबाहुप्रतिबाहुयोगदलघातः ।
अवलम्बपार्श्वभुजयोर्यस्यान्तरमल्पकं तस्य ॥
अन्येषां क्षेत्राणां दूरभ्रष्टं यथा त्रयोदशके ।
त्र्यश्रस्य भुजद्वितये त्रिगुणाष्टभुवः फलं स्थूलम् ॥
षड्युतमध्यर्धशतं सूक्ष्मं षष्टिः प्रजायते यस्मात् ।
सूक्ष्मफलस्यैवाहं साधनकरणानि वक्ष्यामि ॥
i.e., "(It is said that) the product of half the sums of the
sides and counter sides (i.e., the product of half the sum
of the base and face and half the sum of the flank sides)
of a triangle or a quadrilateral, gives the gross value of
the area. But this result is true only for those figures in
which the difference between the altitude and the flank
sides is small. In the case of other figures the above
result is far removed from the truth ; as for example, in
the case of the triangle having 13 for the two (flank) sides
and 24 for the base, the gross value of the area is 156,
whereas the accurate value of the area is 60. I shall,
therefore, state the methods for obtaining the accurate
results only."
The rule which has been so mildly criticised in the
beginning of the above passage is due to Brahmagupta. The
author has very carefully tried to make his remark against
that great writer (Brahmagupta) look like an argument for his
proposal to deal with accurate methods only. But his inten-
tion is clearly understood when we see that the first line of
the passage giving the rule criticised by him is a literal quota-
¹ Rule 112-114.

xx INTRODUCTION
tion from Brahmagupta. [See BrSpSi, xii. 21(i)]. The com-
mentator has made the matter perfectly clear by citing the
whole passage (BrSpSi, xii. 21) from Brahmagupta.
Although Śrīdharâcârya has made Brahmagupta the
main target of his criticism, the corresponding rule of Mahāvîra
(GSS, vii. 7(i)), has also been criticised by implication.
3·4. Indebtedness to Mahavira. The following two
interesting examples, the first occurring in the Pâṭîgaṇita and
the second in the Triśatikâ,
(1) त्रिभिः पारावताः पञ्च पञ्चभिः सप्त सारसाः ।
सप्तभिर्नव हंसाश्च नवभिर्बर्हिणस्त्रयः ॥
राजपुत्रविनोदार्थं ज्ञात्वा मूल्यं यथोदितम् ।
शतेनैकेन रूपाणां जीवानां शतमानय ॥
(PG, Ex. 78-79)
(2) षड्भागः पाटलीषु भ्रमरगणयुतेः स्वत्रिभागः कदम्बे
पादश्चूतद्रुमे च प्रदलितकुसुमे चम्पके पञ्चमांशः ।
प्रोत्फुल्लाम्भोजषण्डे रविकरदलिते त्रिंशदंशोऽभिरेमे
तत्रैको मत्तभृङ्गो भ्रमति नभसि चेत् का भवेद्भृङ्गसङ्ख्या ॥
(Triś, Ex. 28)
are almost literally the same as the following ones occurring
in the Gaṇita-sâra-saṅgraha of Mahāvîra :
(1) त्रिभिः पारावताः पञ्च पञ्चभिस्सप्त सारसाः ।
सप्तभिर्नव हंसाश्च नवभिशिखिनस्त्रयः ॥
क्रीडार्थं नृपपुत्रस्य शतेन शतमानय ।
इत्युक्तः प्रहितः कश्चित् तेन किं कस्य दीयते ॥
(GSS, vi. 152-153)
(2) षड्भागः पाटलीषु भ्रमरवरततेस्तत्त्रिभागः कदम्बे
पादश्चूतद्रुमेषु प्रदलितकुसुमे चम्पके पञ्चमांशः ।
प्रोत्फुल्लाम्भोजषण्डे रविकरदलिते त्रिंशदंशोऽभिरेमे
तत्रैको मत्तभृङ्गो भ्रमति नभसि का तस्य वृन्दस्य सङ्ख्या ।
(GSS, iv. 6)
This shows either that Śrīdharâcârya has borrowed the
above examples from Mahāvîra or that both Śrīdharâcârya

INTRODUCTION xxi
and Mahâvîra have borrowed them from a third common
source. But so long as that third source, if there had really
been one, is not discovered it would not be unnatural to
believe in the first alternative. The possibility that Mahâvîra
might have borrowed from Śrîdharâcârya is ruled out by the
fact that the former was anterior to the latter.
There is another interesting example in the Triśatikâ,
which runs as follows :
कामिन्या हारवल्ल्याः सुरतकलहतो मौक्तिकानां त्रुटित्वा
भूमौ यातस्त्रिभागः शयनतलगणः पञ्चमांशश्च दृष्टः ।
आप्तः षष्ठः सुकेश्या गणक दशमकः संगृहीतः प्रियेण
दृष्टं षट्कञ्च सूत्रे कथय कतिपयैर्मौक्तिकैरेष हारः ॥
(Triś, Ex. 26)
This seems to be inspired by the following example of
the Gaṇita-sâra-saṅgraha, which is typically Mahâvîra's:
काचिद्वसन्तमासे प्रसूनफलगुच्छभारनम्रोद्याने ।
कुसुमासवरसरञ्जितशुककोकिलमधुपमधुरनिस्वननिचिते ॥
हिमकरधवले पृथुले सौधतले सान्द्रचन्द्रमृदुतल्पे ।
फणिफणनितम्बबिम्बा कनकदमलाभरणशोभाङ्गी ॥
पाठीनजठरनयना कठिनस्तनहारनम्रतनुमध्या ।
सह निजपतिना युवती रात्रौ प्रीत्यानुरममाणा ॥
प्रणयकलहे समृध्ये मुक्तामयकण्ठिका तदबलायाः ।
छिन्नावनौ निपतिता तत्त्र्यंशश्चेटिकां प्रापत् ॥
षड्भागः शय्यायामनन्तरानन्तरार्धमितिभागाः ।
षट्संख्यानास्तस्याः सर्वे सर्वत्र सम्पत्तिताः ॥
एकाग्रषष्टिशतयुतसहस्रमुक्ताफलानि दृष्टानि ।
तन्मौक्तिकप्रमाणं प्रकीर्णकं वेत्सि चेत् कथय ॥
(GSS, iv. 17-22)
Note how successfully has Śrîdharâcârya condensed the
matter of six verses into one single verse.
3.5. Popularity of Sridharacarya's Works. The
works of Śrîdharâcârya, being written in simple, clear,
and concise language, were greatly appreciated by the people
in general for whom they were written, with the result that
Indological Truths

xxii INTRODUCTION
the earlier works on the subject which were studied in the
times of Bhâskara I (629 A.D.) and Pṛthûdakasvâmî (860 A.D.)
fell into the background and we do not hear of them any
more. From the tenth century onwards preference was
given to Śrīdharâcârya's works and it is these works that are
found to be generally quoted in astronomical and mathe-
matical literature.
The earliest known quotations from the writings of
Śrīdharâcârya are found to occur in Abhayadeva Sûri's
(1050 A.D.) commentary on the Jaina canonical work, the
Sthânâṅga-sûtra. Abhayadeva does not mention the name of
Śrīdharâcârya but quotes the following portions of certain
verses which are traced to Śrīdharâcârya's Pâṭīgaṇita (Rules
24 and 28) and Triśatikâ (Rules 11 and 15) :
(i) सदृशद्विराशिघातः
(ii) समत्रिराशिसंहतिः
The earliest reference to Śrīdharâcârya by name as well
as quotation from one of his arithmetical works occurs at
two places in the commentary, entitled Sundarî, by Udayadi-
vâkara (1073 A.D.), on the Laghu-Bhâskarîya of Bhâskara I.
The passage quoted at both the places is the rule for reducing
a chain of measures.¹
Other references to Śrīdharâcârya or quotations from
his works are found to occur in the following works :
(1) Bhâskara II's ( 1150 A. D. ) commentary on the
Śiṣya-dhî-vṛddhida of Lalla.²
(2) Bhâskara II's commentary on his own Bîjagaṇita.³
(3) Bhâskara II's commentary on the Siddhânta-śiromaṇi.⁴
¹ The passage quoted is PG, Rule 41, or Triś, Rule 26(ii)-27(i).
² The passages quoted are PG, Rule 33(ii) and Ex. 34.
³ Vide supra, §3.
⁴ See Golâdhyâya, Bhuvanakośa, vs. 52 (comm.).

INTRODUCTION xxiii
(4) Sûryadeva's (b. 1191 A.D.) commentary on the
Âryabhaṭīya.¹
(5) Sûryadeva's commentary on the Laghu-mânasa of
Mañjula (932 A.D.).²
(6) Kâvya-Kalpalatâ-vṛtti by Amara Siṁha Yati (c. 1250
A.D.).³
(7) Siṁhatilaka Sûri's (c. 1275 A.D.) commentary on
the Gaṇita-tilaka of Śrîpati.⁴
(8) Nârâyaṇa's (1356 A.D.) commentary on his Gaṇita-
kaumudî.⁵
(9) Viṣṇu Śarma's commentary (c. 1363 A.D.) on the
Vidyâmâdhavîya.⁶
(10) Makkibhaṭṭa's (c. 1377 A.D.) commentary on the
Siddhânta-śekhara of Śrîpati.⁷
(11) Parameśvara's (1430 A.D.) commentary on the
Âryabhaṭīya.⁸
(12) Râma's (1450 A.D.) commentary on the Kâtyâyana-
śulba.⁹
(13) Yallaya's (1480 A.D.) commentary on the Ârya-
bhaṭīya.¹⁰

¹ The passages quoted are PG, Rules 34 and 35.
² The passages quoted are PG, Rules 32(i), 33(ii), and 43.
³ See Kâvya-kalpalatâ-vṛtti by Amara Siṁha Yati with the text by Ari
Siṁha. Edited by Jagannath Sastri Hoshing, Banaras (1931), p. 111.
⁴ The passages quoted are रूपादिद्विचयपदसमासो वा (PG, Rule 24),
'निर्व्युक्तराशिरन्त्यं तस्य घनोऽसौ (PG, Rule 28), रूपगणच्छेदसङ्गुणः (PG, Rule 39),
and PG, Rule 37. See GT, pp. 4, 9, 11, 17, 31 and 39.
⁵ The passage quoted is PG, Rule 109. See GK, II, p. 49.
⁶ See Vidyâmâdhavîyam by Vidyâmâdhava. Edited with the commen-
tary, entitled Muhûrta-dîpikâ, of Viṣṇu Śarmâ by R. Shamasastry, Part II,
Mysore (1923), p. 249. Part III, Mysore (1926), p. 134. The passage
quoted is PG, Rule 12.
⁷ The passage quoted is PG, vv. 7-8. See SiŚe, I, p. 17.
⁸ The passages quoted are PG, Rules 34(i) and 35(i). See Â, ii. 3
(comm.).
⁹ The passage quoted is Triś, Rule 47. See B. Datta, The Science
of the Śulba, Calcutta (1932), p. 11.
¹⁰ The passages quoted are PG, Rules 27, 32(i) 34(i), 39(i), 40(i),
44(ii), 86(i), and 98 ; and Triś, Rule 47.

xxiv INTRODUCTION
(14) Râghavabhaṭṭa's (1493 A.D.) commentary on the
Śâradâ-tilaka.¹
(15) Gaṇeśadaivajña's (1545 A.D.) commentary on the
Lîlâvatî of Bhâskara .²
(16) Raghunâtharâja's (1597 A.D.) commentary on the
Âryabhaṭîya.³
(17) Raṅganâtha's (1603 A.D.) commentary, entitled
Mitabhâṣiṇî, on the Lîlâvatî of Bhâskara II.⁴
(18) Mahîdhara's (1635 A.D.) commentary on the Lîlâvatî
of Bhâskara II.⁵
(19) Anonymous commentary, entitled the Prayoga-racanâ
on the Mahâ-Bhâskarîya of Bhâskara I.⁶
(20) Râmakṛṣṇadeva's commentary, entitled Manorañjana,
on the Lîlâvatî of Bhâskara II.⁷
(21) Vîreśvara's commentary, called Pâṭîvyâkhyâna or
Pâṭîgaṇita or Lîlâvatyudâharaṇa, on the Lîlâvatî of Bhâskara II.⁸
The above references show that Śrīdharācārya's works
continued to be studied in India till about the middle of the
seventeenth century A.D. Although the great Bhāskara II
wrote the famous Lîlâvatî about the middle of the twelfth
century A.D., he was not successful to oust the works of
Sridharâcârya from the field.
Evidence of the popularity of the Pâṭīgaṇita and the
Triśatikâ is furnished also by the occurrence of commentaries on
those works and by the influence exerted by them on later works.
¹ The passages quoted are PG, Rule 115 and Triś, Rule 45(ii).
² L (ASS), pp. 6, 12, 160, 218. On p. 218, Triś, Rule 47 is quoted.
³ The passages quoted are PG, Rules 34 and 35.
⁴ See Catalogue of the Sanskrit Manuscripts in the Library the India
Office. Part V. Edited by J. Eggeling, London (1896). Catalogue No
2814. Also see N. Ramanujacharia and G. R. Kaye, 'The Triśatikâ of
Śrīdharācārya,' Bibliotheca Mathematica, Vol. XIII (1912-13), p. 205.
⁵ The passages quoted are Triś, Exs. 26 and 28.
⁶ The passage quoted is PG, Rule 39(i). See comm. on MBh, i. 20.
⁷ See N. Ramanujacharia and G. R. Kaye, l.c., p. 205.
⁸ See Catalogue of the Sanskrit Manuscripts in the Library of the India Office.
Part V. Edited by J. Eggeling, London (1895). Catalogue No. 2816.

INTRODUCTION xxv
The Triśatikâ, being more elementary and better sui-
ted to the needs of the people in general, attracted more
readers and seems to have been comparatively more popular,
as is evident from the following facts :
(1) Whereas there is only one manuscript of the Pâṭīgaṇita
available at present, manuscripts of the Triśatikâ are
known to exist at several places in India.¹
(2) Whereas we are aware of only one commentary on
the Pâṭīgaṇita, we know of the following five com-
mentaries on the Triśatikâ:
(i) A commentary by Śambhûnâtha.²
(ii) A commentary by Vṛndâbana Śukla.³
(iii) A commentary by Vallabha in Telugu.⁴
(iv) An anonymous commentary in Kannaḍa.⁵
(v) An anonymous commentary in Gujarâtî.⁶
It is noteworthy that the last-mentioned three
commentaries were written in the regional langu-
ages of India. This is another evidence of the
great popularity of this work.
(3) Manuscripts of the Triśatikâ that are now available
exhibit a remarkable divergence in their texts.

¹ Manuscripts of the Triśatikâ exist in the Central Library, Baroda;
the Oriental Institute, Baroda; the Sanskrit College Library, Banaras;
the Anup Sanskrit Library, Bikaner; the West Punjab University Library,
Lahore; the Library of the India Office, London; the British Museum,
London; the Lucknow University Library, Lucknow; the Jaina Matha,
Moodbidri; the Curator’s Office Library, Trivandrum; and the Scindhia
Oriental Institute, Ujjain.
² Mentioned in the Catalogue of Manuscripts from Guzerat.
³ Mentioned by Dr A. N. Singh.
⁴ A palm-leaf manuscript of this commentary occurs in the Jaina
Matha, Moodbidri. See कन्नडप्रान्तीय ताडपत्रीय ग्रन्थसूची, published by the
Bhâratîya Jñânapîṭha, Kashi, p. 169, MS. No. 781.
⁵ A manuscript of this commentary exists in the Library of the India
Office, London. See Catalogue of Sanskrit and Prâkrit Manuscripts in the
Library of the India Office, Oxford (1935), p. 781, MS. No. 6317.
⁶ A manuscript of this commentary exists in the Oriental Institute,
Baroda, Accession No. 4660. Some extracts from this manuscript have
been quoted by Bhogilal J. Sandesara in his paper, entitled ‘Weights,
Measures and Coinage of Mediaeval Gujrat.’ See Journal of the Numismatic
Society of India, Vol. VIII, Part II, 1946, pp. 138-146.

xxvi INTRODUCTION
This divergence, which is evidently due to continual
modifications and interpolations made by teachers
engaged in teaching this subject, clearly proves that
this work was very popular and continued to be
used as a text-book for a long time.
The works of Śrîdharâcârya were studied not only by
the Hindus but also by the Jainas, as is evident from the
occurrence of references to Śrîdharâcârya or quotations from
his works in the writings of Abhayadeva Sûri, Siṁhatilaka
Sûri, and Amara Siṁha Yati, who were all Jaina scholars.
Vallabha, who wrote a commentary on the Triśatikâ in the
Telugu language, was also a Jaina.
Appreciation of Śrīdharâcârya.
The works of Śrîdharâcârya, which were so popular,
earned a great name for their author. The following verse,
which is generally cited in appreciation of him both by the
Hindus and the Jainas gives an idea of the unique position
occupied by him as a mathematician :
उत्तरतो सुरनिलयं दक्षिणतो मलयपर्वतं यावत् ।
प्रागपरोदधिमध्ये नो गणकः श्रीधरादन्यः ॥¹
“Up to the abode of the gods (i.e., the Himalayas)
towards the north and up to the Malaya mountain
towards the south and between the eastern and western
oceans, there is no mathematician except Śrîdhara.”
From the Kâvyakalpalatâ-vṛtti² of Amar Siṁha Yati we
learn that Śrîdharâcârya was considered to be so great as a
mathematician that the ideal of a Hindu mathematician was
to be a Śrîdhara.
3·6. Influence on later writers.
(1) Influence on Âryabhaṭa II (c. 950 A.D.). We have
already seen that the rules for (i) reducing a chain of
measures, (ii) finding the area of a segment of a circle and
(iii) finding the volume of a sphere, occurring in the Mahâ-
――――――――――――――――――――――――――――――――
¹ This verse occurs as a quotation in the Gujarâtî commentary,
referred to above. It is also found to occur in the end of several manu-
scripts of the Triśatikâ, including that containing the commentary by
Vallabha.
² See Kâvyakalpalatâ-vṛtti, p. 111.

, xv. 17(ii))"
Then:
"अंशघनमूलराशौ छेदनमूलोद्धृते मूलम् ।"
And below it, aligned to the right: "(Triś, Rule 22(ii))"
Then:
"(v) बालेन्दौ त्रिभुजे द्वे"
And below it, aligned to the right: "(MSi, xv. 101(i))"
Then:
"बालेन्दौ त्रिभुजे द्वे"
And below it, aligned to the right: "(Triś, Rule 44(ii))"

Everything is very consistent!
Let's format each line cleanly and faithfully.
No LaTeX math syntax. Pure markdown with line markers , , etc.INTRODUCTION xxvii
siddhânta, are due to the influence of Śrîdharâcârya. There
is, in general, a close agreement between the contents of the
chapter on pâtîgaṇita of the Mahâ-siddhânta and the Triśatikâ,
and it appears that Âryabhaṭa II, while writing that chapter,
made a constant consultation of the Triśatikâ. The following
similarities in language between the two works deserve special
notice :

xxviii INTRODUCTION
(ii) यूथार्धं सत्रिभागं गिरिवरशिखरे कुञ्जराणां प्रणष्टं
षड्भागश्चापि नद्यां पिबति च सलिलं सप्तमांशेन युक्तः ।
पद्मिन्यामष्टमांशः स्वनवकसहितः क्रीडते पद्मषण्डे
नागेन्द्रो हस्तिनीभिस्तिसृभिरनुगतः का भवेद् यूथसंख्या ॥
(GT, p. 42, lines 9-12)¹
(iii) शतस्य भाव्यके यत्र षड् भवन्ति पृथक् सखे ।
तत्र रूपसहस्रस्य मध्यतः किं भवेद् वद ॥
(GT, p. 70, lines 11-12)²
(iv) द्विकव्यासोऽष्टकायामः कम्बलो लभते दश ।
अन्यौ द्वौ त्रितवायामौ किमाप्नुतः कथ्यताम् ।
(GT, p. 78, lines 23-24)³
(v) आयामव्यासपिण्डेन नवपञ्चैकहस्तिका ।
लभतेऽष्टौ शिलान्ये द्वे दशसप्तद्विहस्तिके ॥
(GT, p. 79, lines 9-10)⁴
A number of other rules and examples occurring in the
Gaṇita-tilaka are similar to those found in the Triśatikâ, and
seem to be based on them.⁵
(3) Influence on Bhâskara II (1150 A.D.) and Nârâyaṇa
(1356 A.D.). Influence of Śrīdharâcârya’s works is also visible
in the Lîlâvatî of Bhâskara II and the Gaṇita-kaumudî of
Nârâyaṇa.
4. Present Commentary on the Patiganita. The
commentary subjoined to the text of the Pâṭîgaṇita in the
present edition is entitled ṭîkâ. It aims at explaining the
¹ Cf. Triś, Ex. 27.
² Cf. Triś, Ex. 37.
³ Cf. Triś, Ex. 51.
⁴ Cf. Triś, Ex. 52.
⁵ For example, compare :—
GT, p. 39, lines 7-10 with Triś, Rule 26(ii)-27(i).
GT, p 74, lines 8-9 with Triś, Ex. 38.
GT, p. 74, line 16-17 with Triś, Ex. 47.
GT, p. 75, lines 6-9 with Triś, Ex. 44.
GT, p. 77, lines 22-25 with Triś, Ex. 50.
GT, p. 78, lines 8-11 with PG. Ex. 49.
GT, p. 82, lines 2-5 with Triś, Ex. 51.
GT, p. 83, lines 5-6 with Triś, Ex. 57.
GT, p. 83, lines 25-28 with Triś, Ex. 59.
GT, p. 86, lines 25-28 and p. 87, lines 1-2, 22-23 with Triś, Exs.
60-62.

INTRODUCTION xxix
meaning of the text without going into unnecessary details.
Under each rule the commentator explains and discusses the
rule and throws light on other relevant matter connected
with the text. Under each example he first paraphrases the
example, explaining it where necessary, and then solves it by
closely following the steps of the rule. At some places the
commentator quotes from earlier writers but on no occasion
does he go out of point. The language used in the commen-
tary is mature and the style of expression is generally that
obtaining in philosophical works.
The name of the commentator does not occur anywhere
in the available commentary. There is also no reference to
the place to which he belonged or to the time when the com-
mentary was written. The following facts, however, seem to
indicate that the commentator belonged to some place in the
north-west region of India, probably to Kashmir :
(i) The manuscript of the present commentary is written
in the Kashmîrî Script and was discovered in Kashmi
(ii) The symbols + and the dot, which the commentator
uses for expressing the negative sign and the zero
respectively, are the same as those used in the com-
mentary of the Bakhshâlî text, which was discovered
in the north-west region of India, but different from
those used by the mathematicians belonging to other
regions of India.
(iii) The hemistich तयोर्योगे वियोगः स्याद् वियोगे सति सङ्गमः , which
the commentator cites from some unknown work is
also found to be cited in the commentary of the
Bakhshâlî text.
(iv) The passages cited as quotations by the commenta-
tor are generally not to be found in the writings of
the other mathematicians or astronomers who hail
from the other parts of India.
The commentator is also silent about the sources from
which he has cited passages as quotations in the commentary

xxx INTRODUCTION
We have however, been, able to trace the following two
quotations to the Brâhma-sphuṭa-siddhânta of Brahmagupta :
(i) पदमेकहीनमुत्तरगुणितं संयुक्तमादिनाऽन्त्यधनम् ।
आदियुतान्त्यधनार्धं मध्यधनं पदगुणं गणितम् ॥ ¹
(ii) स्थूलफलं त्रिचतुर्भुजबाहुप्रतिबाहुयोगदलघातः ।
भुजयोगार्धचतुष्टयभुजोनघातात् पदं सूक्ष्मम् ॥ ²
The following two quotations, in the âryâ metre, though
wrongly ascribed by the commentator to Śrîdharâcârya, are
also traced to the Brâhma-sphuṭa-siddhânta of Brahmagupta :
(iii) धनयोर्धनमृणमृणयोर्धनर्णयोरन्तरं समैक्यं खम् ।
ऋणैक्यमृणं धनशून्ययोर्धनं शून्ययोः शून्यम् ॥ ³
(iv) ऊनमधिकाद्विशोध्यं धनं धनादृणमृणादधिकमूनात् ।
व्यस्तं तदन्तरं खादृणं धनं धनमृणं भवति ॥
शून्यविहीनमृणमृणं धनं (धनं) भवति शून्यमाकाशात् ।
शोध्यं यदा धनमृणादृणं धनाद्वा तदा क्षेप्यम् ॥ ⁴
(v) ऋणमृणधनयोर्घातो धनमृणमोर्धनवधो धनं भवति ।
शून्यर्णयोः खधनयोः खशून्ययोर्वा वधः शून्यम् ॥ ⁵
(vi) धनभक्तं धनमृणहृतमृणं धनं (भवति) खं खभक्तं खम् ।
भक्तमृणेन धनमृणं धनेन हृतमृणमृणं भवति ॥
खोद्धतमृणं धनं वा तच्छेदः खमृणधनविभक्तं खम् । ⁶
(vii) धनमृणधनयोर्वर्गः खं खस्य पदं कृतिर्यत्तत् ॥ ⁷
The sources of the following quotations remain untraced:
(i) ग्राह्यमूलोऽथ मूलश्च करणीसंज्ञितावुभौ ।
(p. 8, lines 6)
(ii) रूपे भाज्यसमो लाभः ऊनो रूपाधिके भवेत् ।
अधिको रूपरिक्ते तु तुल्ये रूपं हि नीयते ॥
(p. 32, lines 10–11)

¹ p. 118-119, and p. 137. Cf. BrSpSi, xii. 17.
² p. 159, lines 22-23. ” ” xii. 21.
³ p. 123, lines 22-23. ” ” xviii. 30 (31).
⁴ p. 124, lines 2-5. ” ” xviii. 31-32 (32-33).
⁵ p. 124, lines 15-16. ” ” xviii. 33 (34).
⁶ p. 124, lines 21-23. ” ” xviii. 34-35(i) [35-36(i)].
⁷ p. 125, line 2. ” ” xviii. 35(ii) [36(ii)].

INTRODUCTION xxi x
(iii) वधे प्रवर्तिते रूपाद्धि भागे तल्लीयते फलम् ।
आद्ये न्यूनाधिके सङ्गेत् विपरीतं तथोत्तरे ॥
(p. 32, lines 14–15)
(iv) त्रिशन्नवाधिका रूपं रूपं चेति क्रमाद् गुणाः ।
आद्यो द्व्यूनोत्तरो मध्य एकवृद्ध्या परः स्थिरः ॥
आद्या नवोना मध्यास्तु प्राग्वदन्याः सरूपकाः ।
तथा—एकविंशतिराद्यस्तु रूपमेकं त्रयस्तथा ॥
आद्यो द्व्यूनो मध्य एकवृद्ध्याऽन्यः स्थिर एव च ।।
यावद् दश-द्वि-चत्वारः सर्वेऽमो खर्णकारिणः ।।
अष्टौ द्वयुत्तरसंशुद्धास्त्रयो रूपोत्तराधिकाः ।
चत्वार एवोत्पाद्यास्तु गुणकाः शेषकारिणः ।।
(p. 82 lines 8-15)
(v) अन्योनात्तेन गुणिताँत् तत्र नाप्ताधिको गुणः ।
गुणितागुणितान्योनामेवमेव विचिन्तयेत् ॥
(p. 82, lines 15-16)
(vi) स्याद् योगे वियदूनेभ्यो वियोगे तद्विपर्ययः ।
(p. 111, line 13 ; p. 114, lines 12)
(vii) तयोर्योगे वियोगः स्याद् वियोगे सति सङ्गमः ।
(p. 111, lines 17 ; p. 132, lines 4 and 6)
(viii) सदृशा(दृ) द्विगतक्षिवत् ।
(p. 121, lines 21)
(ix) ताभ्यां व्यक्तगुणाहते ।
(p. 121, lines 22)
(x) रूपाणां च कृते कार्ये पक्षयोरुचापवर्तने ।
अस्वपक्षे चतुर्वर्गहतेऽव्यक्तकृतौ युते ।।
मूलं व्यव्यक्तमघोनं प्रमाणं वर्गभाजितम् ।
(p. 122, lines 1–3)
(xi) संशोध्याव्यक्तमेकस्मात् पक्षाद्रूपाणि चान्यतः ।
रूपशिष्टप्रमाणं स्यान्निच्छिष्टाव्यक्तस्य तत् फलम् ।।
(p. 126, lines 4–5)
(xii) यद्वा व्यक्तहते वर्गेऽव्यक्तार्धकृतिना युतम् ।
मूलं तेनोनमव्यक्तप्रमाणं वर्गभाजितम् ।।
(p. 133, lines 18–19)

lxxii INTRODUCTION
(xiii) भुजस्य कृतिरिष्टस्य भक्तोनेष्टेन तद्दलं ।
कोटिरिष्टाधिका कर्णश्चतुरश्रायतस्य ते ॥
(p. 159, lines 4-5.)
(xiv) गुणके वर्गयोर्मध्ये तत्पदाधो भुजश्रुती ।
केचित्प्राक्कथिते तत्र वज्रकेणाहती तयोः ॥
अन्तरस्य कृतिः क्षेपस्तत्कोटिः प्रथमं पदम् ।
ऋजुहत्यन्तरं ज्येष्ठं रूपक्षेपेन्तरोद्धृते ॥ (p. 159, lines 9-12)
Quotation (xiv) above deserves special attention. It
gives the rational solution of the equation Nx² + 1 = y² in the
form¹
k Ah − Bb
x = ─────── , y = ───────── ,
Bh ~ Ab Bh ~ Ab
where b, k, and h are respectively the base, upright, and
hypotenuse of a right-angled triangle (which are determined
by the rule given in quotation xiii), A, B, being two numbers
such that A² − B² = N.
This form of solution of Nx² + 1 = y² is unique in so far as
it does not find its parallel in any other work on mathematics,
ancient or modern. It is at the same time the most general
rational solution of that equation. The solutions given by
other mathematicians, such as Brahmagupta (628 A.D.),
Śrîpati (1039 A.D.), Bhâskara II (1150 A.D.), Nârâyaṇa (1356
A.D.), Jñânarâja (1503 A.D.), and Kamalâkara (1658 A.D.)
and also those given by John Wallis and W. Brouncker (1657
A.D.) are easily deducible from it.
The hemistich of quotation (vii) is also of great interest.
For, as we have already mentioned, it is found to occur as a
quotation in the Bakhshâlî Manuscript also. There are two
places in the Bakhshâlî Manuscript where it is seen to be quoted.
At one place we have
तयोयोगवियो......... (BM, 54 recto)
─────────────────────────────────────────────────────────────
¹ For details, see my paper, entitled 'On Śrîdhara's Rational Solution
of Nx² + 1 = y²', Gaṇita Vol. I, No. 2, pp. 1-12. [It must be noted that the
above rational solution is due to some unknown mathematician, who
was the author of the verses of quotation (xiv), and not to Śrîdharâcârya
as was supposed in that paper.]

INTRODUCTION xxxiii
and at the other place, we have
कस्मात् कारणा । तयोर्योगवियोगस्यावियोगस·······
(BM, 35 recto)
Although the quotation occurs in a mutilated form in both
the places, but there is no doubt about the line quoted.
The occurrence of the above hemistich in the Bakhshālî
Manuscript as well as in the present commentary seems to
suggest that the work from which it has been taken was quite
popular in the north-west region of India.
Amongst the notable features of the commentary may
be mentioned (1) the use of the symbol + for the minus sign,
and (2) the use of a dot as well as a small circle for the zero.
The symbol + is generally used after the number affected, but
sometimes it is also found to be used before the number
affected. Thus, - 2 is written as 2+ , and sometimes as +2.
The symbols + and the dot are known to have been used in
the same sense in the Bakhshālî Manuscript also. Their occur-
rence in the Bakhshālî Manuscript and the present commen-
tary seems to suggest that they were in general use in the
north-west division of India.
The dot as a symbol for the zero is believed to be more
ancient than the small circle. The use of both these symbols
in the present commentary shows that the mathematicians in
the north-west region of India had not totally given up the use
of a dot and were using it as an alternative symbol for the zero.
In other parts of India the use of the dot was totally given up
and do we not find it used in any other work written after the
seventh century A.D. Absence of the use of a small circle as
a symbol for the zero in the Bakhshālî Manuscript shows that
this work is much older than the present commentary.
There are five places in the present commentary where
the scribe of the manuscript has made certain interpolations
by way of filling up the apparent gaps in the commentary.
In all these places, however, he has clearly stated what inter-
polations he has made.

xxxiv INTRODUCTION
The first interpolation is the entire commentary to Ex. 70,
at the end of which the interpolator says :
एष च द्वितीयोदाहरणश्लोकष्टीकाकृता स्वटीकायां न व्याख्यातः।
i.e., "This verse of the second example was not explained
by the commentator in his ṭîkâ."¹
The second interpolation comprises lines 19-23 on page
119, the third lines 17-20 on page 120, and the fourth lines
11 16 on page 123. In regard to these three interpolations, the
interpolator makes the following general remark on p. 123 :
अस्य चतुर्थोदाहरणस्य 'आदिः पदहृतगणितमि'त्यतः प्रभृत्युदाहरणेषु टीकाकृता
त्रयाणामुदाहरणानामाद्याद्यानयनप्रकारे दर्शिते भूय आद्याद्यानयनप्रकारो न दर्शित इति
वृत्तिग्रन्थे मूलग्रन्थे चास्योदाहरणस्य दृष्टत्वादस्माभिर्लेखनावसर एव तदानयनप्रकारो दर्शित
इत्यलम् ।
i.e., "The commentator (ṭîkâkâra) having explained the me-
thod of finding the first term etc. in relation to the first three
examples given after the rule 'âdiḥ padahṛtagaṇitam', etc., has
not explained the method of finding the first term, etc., again
in relation to the fourth example. But, this fourth example be-
ing available in 'the book containing the text' as well as in
'the book containing the commentary,' we have ourselves
explained that method while making this copy."
The fifth interpolation occurs on p. 167 and comprises
the lines occurring after the following remark of the interpo-
lator :
यथा चात्र पृथक् खण्डत्रयात् क्षेत्रफलानयनं तथा चाग्रिमोदाहरणे टीकाकृदेव
प्रकटयिष्यतीत्यलम् ।
i.e., "Just as here the area is (being) obtained by deform-
ing the whole figure into three parts, so in the next example
the commentator himself will explain."
5. SRIDHARACARYA. Having thus briefly noticed
the Pâṭîgaṇita and the present commentary on it, let us now say
something regarding Śrîdharâcârya, the author of the Pâṭîgaṇita
The personal history of this author is shrowded with great obs-
curity. The author mentions his name in the opening stanza

INTRODUCTION xxxv
of the Pāṭīgaṇita-sâra, but nowhere does he give out the name
of his father, mother, or teacher, or the time and place of his
birth and literary activity. Later writers, too, have thrown
little light on these aspects of his life. The determination of his
time and place is, therefore, a very difficult problem. In the
following sections we have endeavoured to ascertain as much
about his life as could be possible on the basis of circumstantial
evidence furnished by his works or by a comparative study of
his works with those of other writers of known time and
place.
5·1. A Saivite Hindu. In the opening stanza of the
Pātīgaṇita, Śrīdharâcârya pays homage to the Unborn God, the
cause of creation, preservation, and maintenance of the worlds.
The commentator identifies this Unborn God with God
Maheśvara or Śiva, who is endowed with five functions
(pañcakṛtya), viz., creation (sṛṣṭi), preservation (sthiti), destruc-
tion (saṁhâra), kindness (anugraha), and disappearance (tirodhâ-
na).¹ In the opening stanza of the Pātīgaṇita-sâra, Śrīdhar-
âcârya pays homage to God Śiva by name. Hence it is clear
that he was a devotee of God Śiva. This conclusion is
furthermore corroborated by the inclusion in the Pātīgaṇita of
an example which relates to the worship of the five faces of
God Śiva. (See Ex 89-90). This, it may be added, is the
only example in the works of Śrīdharâcârya that has reference
to a god.
5·11. Not a Jaina. N. C. Jaina says that in the
opening stanza of the Triśatikā, in a palm-leaf manuscript in
Kannaḍa, discovered in a Jaina library in Karnatak,
there is found to occur the word jinam in place of śivaṁ, and
so he thinks that Śrīdharâcârya was a Jaina and not a Saivite
¹ Cf. Âryavidyâsudhâkara by Yajñeśvara Cimaṇa Bhatta, edited with
notes, etc., by MM. Pandit Śivadatta D. Kudâla, published by Motilal
Banarsidas, Lahore (1923), p. 131, where we have the following verse
giving the five functions of God Mahaśvara :
पञ्चविधं तत्कृत्यं सृष्टिस्थितिसंहारतिरोभावाः ।
तद्वदनुग्रहकरणं प्रोक्तं सततोदितस्यास्य ।