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

INTRODUCTION vii
Professors Maskarî Pûraṇa, Mudgala, and others. How can
that be stated by the Âcârya (Âryabhaṭa) in a small work (like
the Âryabhaṭîya)? So we have rightly said ‘a bit of mathema-
tics.’”¹ The works of Maskarî Pûraṇa and Mudgala referred
to above have not survived the ravages of time, but from what
Bhâskara I says about them they must have been exclusively
devoted to pâṭīgaṇita and algebra. Some of them might have
existed before the time of Âryabhaṭa I. Bhâskara I has
quoted a numbers of arithmetical rules from those works,
which tend to show that early works on arithmetic, though
generally of the same nature as later works, now available to
us, had the special peculiarity of including rules for verifying
the results of calculation. Those who have had the opportu-
nity of going through the Bakhshâlî Manuscript must have
noted that varification forms an intrinsic part of solutions given
in that manuscript. In the Âryabhaṭîya, too, we find a rule
meant for the verification of areas of plane figures. (See
Â, ii. 9(i)). Bhâskara I has also sometimes verified his solutions.
In later works we seldom come across any rules dealing with
the verification of answers.
There is another important Hindu mathematician,
Skandasena by name, who lived prior to the ninth century
A.D. Pṛthûdakasvâmî (860 A.D.) has referred to this
mathematician three times in his commentary on Chapter XII
of the Brâhma-sphuṭa-siddhânta of Brahmagupta. He has made
the following remarks in connection with that mathematician :
(1) “The sum (of the series) which Âcârya Skandasena
has interpreted with the help of a series-figure (śreḍhî) is meant
to illustrate the figure (of the series).”²
(2) “The Âcârya (Brahmagupta) has stated here only
five varieties (of fractions). The sixth variety has been omitted,
as it consists of the rest and is therefore virtually taught. It

¹ Bhâskara I’s comm. on Â, i. 1.
² यच्च स्कन्दसेनाचार्येण श्रेढीन्यायेन सङ्कलितं प्रदर्शितं तत् सङ्कलनं क्षेत्रप्रदर्शनाय ।
See Pṛthûdakasvâmî’s comm. on BrSpSi, xii. 2.

viii INTRODUCTION
has been given by Skandasena and others under the name of
bhâgamâtâ.”¹
(3) “This method (of multiplication) by parts is taught
by Skandasena and others. In like manner the other methods
of multiplication such as tatstha and kapâṭa-sandhi, taught by the
same authors, may be inferred by the students’ own ingenuity.”²
These references to Skandasena are of great historical
importance, as they show that the methods of multiplication
known as kapâṭa-sandhi (‘door-junction method’), tatstha, and
khaṇḍa (‘multiplication by parts’), as well as the mixed fractions
called bhâgamâtâ occurred also in early works on Hindu pâṭî-
gaṇita. We also learn that the geometrical interpretation of an
arithmetic series with the help of a series-figure which has been
explained in detail in Śrîdharâcârya’s Pâṭîgaṇita (See Rules
79-84) is due to Skandasena or some earlier mathematician
and not to Śrîdharâcârya.
The works of the mathematicians Skandasena and others
mentioned by Pṛthûdakasvâmî are lost, and we are not in a
position to say anything regarding them with definiteness.
We have, however, recently discovered the following lines in a
palm-leaf manuscript of a work on mathematics, entitled
Gaṇitâvalî, which indicate that the works of Skandasena and
other early writers were generally found to be difficult and
unintelligible by the people :
स्कन्दसेनादिभिर्यानि कृतानि निखिलं पुरा ।
अतिगूढानि शास्त्राणि दुर्बोधानि बहु .. .... ॥
व्यवहारो न तैः सर्वैरधुना दृश्यते क्वचित् ।
तस्माद्विहाय·दुर्बोधान् विधीन् चाव्यवहारिकान् ॥
व्यवहारोऽद्यापि लोके तान् वक्ष्याम्यनुपूर्वशः ।³
¹ एवमिहाचार्येण पञ्चजातय एवोक्ताः । यत षष्ठीति तदात्मकैवातो गतार्थेति कृत्वा
नोक्ता । स्कन्दसेनादिभिस्तस्या नाम कृतं भागमातेति । See Pṛthûdakasvâmî’s comm.
on BrSpSi, xii. 9
² See Pṛthûdakasvâmî’s comm. on BrSpSi, xii. 55.
³ See A Descriptive Catalogue of the Sanskrit Manuscripts in the Collections
of the Royal Asiatic Society of Bengal, Calcutta (1945), p. 80, Ms. No. 6924,
Gaṇitâvalî, verses 9-11 (i).

INTRODUCTION ix
i.e., “The works which were written in the past by Skandasena
and others are very difficult and unintelligible and are
nowhere seen to be in practice now. Therefore, leaving aside
the difficult and out-of-use methods, I shall say the methods
which are in practice even now in their proper sequence.”
The following works written during the eighth and
ninth centuries are also now lost :
(1) The Pâtîgaṇita of Lalla (8th century A.D.)
Quotations from this work occur in Bhâskara II’s
commentary on the Siddhânta-śiromaṇi¹ and in the
Gaṇita-Kaumudî² of Nârâyaṇa (1356 A.D.).
(2) The Siddhânta-tilaka of Lalla (8th century A.D.)
This work was similar to the Brâhma-sphuṭa-siddhânta
of Brahmagupta, and contained chapters on
arithmetic and algebra. Quotations from this
work occur in the commentaries of Râmakṛṣṇa
Ârâdhya (1472 A.D.) and Yallaya (1480 A.D.)
on the Sûrya-siddhânta.
(3) The Govindakṛti of Govinda (9th century A.D.)
This work, written as a sequel to the Âryabhaṭîya of
Âryabhaṭa I, was also similar to the Brâhma-sphuṭa-
siddhânta of Brahmagupta, and contained chapters
on arithmetic and mensuration. Quotations from
this work are found to occur in the commentaries of
Śaṅkaranârâyaṇa (869 A.D.) and Udayadivâkara
(1073 A.D.) on the Laghu-Bhâskarîya of Bhâskara
I, and in Nîlakaṇṭha’s commentary on the Ârya-
bhaṭîya. The author Govinda was a teacher of
Śaṅkaranârâyaṇa.

¹ Bhuvanakośa, vv. 54-57 (comm.)
² II, p. 175.

x INTRODUCTION
At present we are in possession of the following works
on arithmetic and mensuration :
(1) The Gaṇita-sâra-saṅgraha of Mahâvîra (850 A.D.)
(2) The Pâṭīgaṇita and the Pâṭīgaṇita-sâra (or Triśatikâ)
of Śrîdharâcârya.¹
(3) The Gaṇita-tilaka of Śrîpati (1039 A.D.). Incomplete.
(4) The Lîlâvatî of Bhâskara II (1150 A.D.)
(5) The Gaṇita-kaumudî of Nârâyaṇa (1356 A.D.)
We also possess the following astronomical works
which deal with arithmetic and mensuration in one of the
chapters:
(1) The Brâhma-sphuṭa-siddhânta of Brahmagupta (628
A.D.), of which the twelfth chapter, entitled Gaṇitâdhyâya,
deals with arithmetic and mensuration.
(2) The Mahâ-siddhânta of Âryabhaṭa II (c. 950 A.D.),
of which the fifteenth chapter, entitled Pâṭīgaṇita, is devoted
to arithmetic and mensuration.
(3) The Siddhânta-śekhara of Śrîpati (1039 A.D.), of which
the thirteenth chapter, entitled Vyakta-gaṇitâdhyâya, is devoted
to arithmetic and mensuration.
The above works (or chapters) are broadly divided into
two sections, logistics (parikarma) and determinations (vyava-
hâra). The logistics comprise the fundamental operations of
arithmetic and other important methods or processes of calcu-
lation. The determinations relate to topics of commercial
or general interest. In the case of each topic we have a rule
or a set of rules followed by one or more examples. In astro-
nomical works which partly deal with the subject, only rules
are given, and there are no examples.

¹ We have shown later that Śrîdharâcârya lived sometime between
Mahâvîra (850 A.D.) and Âryabhaṭa II (c. 950 A.D.)

INTRODUCTION xi
According to Brahmagupta there are twenty logistics
and eight determinations. He says :
"He who distinctly and severally knows the twenty
logistics, addition, etc., and the eight determinations ending
with shadow, is a mathematician."¹
The twenty logistics, according to Brahmagupta's com-
mentator, Pṛthûdakasvâmî, are : (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), (9-13)
pañca-jâtayaḥ (five varieties of reduction of fractions), (14)
trairâśika (rule of three), (15) vyasta-trairâśika (inverse rule of
three), (16) pañca-râśika (rule of five), (17) sapta-râśika (rule
of seven), (18) nava-râśika (rule of nine), (19) ekâdaśa-râśika
(rule of eleven), and (20) bhâṇḍa-pratibhâṇḍa (barter of com-
modities). The eight determinations are : (1) miśraka
(mixtures), (2) śreḍhî (series), (3) kṣetra (plane figures), (4)
khâta (excavations), (5) citi (piles of bricks), (6) krâkacika
(saw problems), (7) râśi (heaps or mounds of grain), and (8)
châyâ (shadow).
More or less the same topics have been recognized by
the other writers also.
3. WORKS OF SRIDHARACARYA. Śrîdharâcârya
is the only author who wrote two works dealing exclusively
with arithmetic and mensuration, viz., the Pâṭîgaṇita and the
Pâṭîgaṇita-sâra (‘Abstract of the Pâṭîgaṇita’). These names
were given to those works by the author himself. For in the
opening stanza of the former, he says :
"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."
And in the opening stanza of the latter, he says :
¹ BrSpSi, xii. 1.

xii INTRODUCTION
“Śrîdharâcârya, having bowed to God Śiva, tells the
essence of gaṇita as extracted from the Pâṭî(gaṇita) composed
by himself for the use of the people.”
Later writers have sometimes called the Pâṭīgaṇita by the
name Bṛhatpâṭî (‘Bigger Pâṭî’, or ‘Bigger Work on Pâṭīgaṇita’)
to distinguish it from the smaller work of the author. Thus
Râghavabhaṭṭa (1493 A.D.) in his commentary, entitled
Padârthâdarśa, on the Śâradâtilaka, referring to the two works
of Śrîdharâcârya, writes :
अत्रापि सूक्ष्मगणनया अङ्गुलत्रयं न्यूनम्। यतस्तेन ‘व्यासार्धवर्गवर्गात् क्षेत्रफलं
दशगुणान्मूलम्’ति श्रीधराचार्योक्तेन प्रकारेण क्षेत्रफलमानीतं तच्च स्थूलम्। यतो गणित-
ग्रंथानामियं शैली यत् स्थूलं सूक्ष्ममपि फलानयनमुच्यते । तत्र भगवता श्रीधराचार्येण ,
बृहत्पाट्यां प्रकारद्वयमप्युक्त्वा तत्संग्रहे त्रिशतीग्रन्थे स्थूला एव प्रकारा दर्शिताः। भास्कराचार्येण
तु लीलावत्यां स्थूला इव सूक्ष्मा अपि प्रकारा उक्ताः। तां बृहत्पाटीं चावलोक्य मया
सूक्ष्मं फलमिदमानीतम्। तदेव सूक्ष्ममिति मन्यते चेत्तदा मूले अत्यन्तं फलाधिक्यं स्यात् ।¹
Some writers on the other hand, have called the Pâṭî-
gaṇita by the name of Navaśatî (‘A collection of 900 verses’),
which shows that this work comprised about 900 verses. The
earliest reference to this name occurs in Makkibhaṭṭa’s com-
mentary (1377 A.D.) on the Siddhânta-śekhara of Śrîpati.
In contradistinction to the name Navaśatî given to the
bigger work, the smaller work (viz. Pâṭīgaṇita-sâra) is called by
later writers by the name Triśatî or Triśatikâ (‘A collection
of 300 verses’). The name Triśatî occurs in the commentary
by Siṁhatilaka Sûri (1215 A.D.) on the Gaṇita-tilaka of Śrîpati,²
in the commentary by Râghavabhaṭṭa (1493 A.D.) on the
Śâradâtilaka³, and in the colophons to some manuscripts of

¹ See Śaradâtilakam by Lakṣmaṇadeśikendra, with the Padârthâdarśa
commentary by Râghavabhaṭṭa. Edited by Mukunda Jha Bakshi, Banaras
(1934), pages 98-99 (comm.).
² Gaṇita-tilaka by Śrîpati, with the commentary of Siṁhatilaka Sûri,
edited by H. R. Kâpadîâ, Baroda Oriental Institute (1937), pp. 4, 9, 11,
17, and 39.
³ See the passage quoted above.

INTRODUCTION xiii
that work. The name Triśatikā occurs in the colophon to a
manuscript occurring in the Central Library, Baroda.
The Pāṭīgaṇita-sâra has also been called by the abridged
names Pāṭî-sâra or Gaṇita-sâra. But these names are generally
found in the manuscripts of that work.¹
Besides the two works mentioned above, one more work,
dealing with algebra, is attributed to Śrîdharâcârya. The
celebrated Bhâskarâcârya², the author of the Lîlâvatî, while
commenting on his Bījagaṇita, quotes the following rule, deal-
ing with the solution of the quadratic equation, from the
algebraical work of Śrîdharâcârya :³
चतुराहतवर्गसमै रूपैः पक्षद्वयं गुणयेत् ।
अव्यक्तवर्ग रूपैर्युक्तौ पक्षौ ततो मूलम् ॥
i.e., "multiply both the sides (of the quadratic equation
ax²+bx=c) by a known quantity equal to four times the co-
efficient of the square of the unknown ; add to both sides a
known quantity equal to the square of the (original) coefficient
of the unknown ; then extract the square root."
In a concluding stanza to his Bījagaṇita, Bhâskarâcârya
says :
"As the works on algebra ‘of Brahmagupta, Śrîdhara,
and Padmanâbha are very extensive, so for the satisfaction of

¹ According to S. B. Dîkṣita, reference to Gaṇitasâra is made by
Narapati (1175 A.D.) in his Narapati-jayacaryâ. See S. B. Dîkṣita, Bhâratîya
Jyotiṣaśâstra (in Marathi), Second Edition, Poona (1931), p. 476.
² Also called Bhâskara II.
³ See Bhâskara II's Bījagaṇita (Banaras Sanskrit Series), p. 61.
This quotation occurs also in the Bījagaṇita of Jñānarâja (1503) and
in Sûryadâsa's (1541) commentary on Bhâskara II's Bījagaṇita.
Kṛṣṇa (c. 1580) and Râmakṛṣṇa (c. 1648), have also quoted the
above verse in their commentaries on Bhâskara II's Bījagaṇita. But
according to them, the second line of the verse should be
पूर्वव्यक्तस्य कृतेः समरूपाणि क्षिपेत्तयोरेव ॥
i.e., "add to them known quantities equal to the square of the original
coefficient of the unknown."
See B. Datta and A. N. Singh, History of Hindu Mathematics, Part II,
Lahore (1938), p. 65, footnote.

xiv INTRODUCTION
the students I have taken the essence of those works and com-
piled this small work with demonstrations."¹
This shows that Śrîdharâcârya's Bîjagaṇita was an ex-
tensive work on the subject.
Other works ascribed to Śrîdharâcârya
Sudhâkara Dvivedî² thought that Śrîdharâcârya was the
same person as the author of the philosophical work, the Nyâya-
kandalî (written in 991 A.D.), and of the astrological work, the
Jâtaka-paddhati³. S. B. Dîkṣita⁴ also took him to be the same
person as the author of the Jâtaka-paddhati. Unfortunately
there is no definite evidence to identify Śrîdharâcârya with
the author of either of the above two works. There is, how-
ever, some likelihood that the authors of the Triśatikâ and the
Jâtaka-Paddhati were one and the same person. For Viṣṇu
Śarmâ⁵ (c. 1363 A.D.), in his commentary on the Vidyâmâ-
dhavîya, while quoting from the Triśatikâ and the Jâtaka-
Paddhati, does not differentiate between the authors of the two
works, and refers to both of them by the name of Śrîdhara.
N. C. Jaina⁶, presuming that Śrîdharâcârya was a Jaina,
identified him with the authors of the following works :
(i) Jâtaka-tilaka (in Kannaḍa), written in 1049 A.D.
(ii) Lîlâvatî (in Kannaḍa), written after 1150 A.D.
¹ ब्राह्माह्वयश्रीधरपद्मनाभबीजानि यस्मादतिविस्तृतानि ।
आदाय तत्सारमकारि नूनं सयुक्तियुक्तं लघु शिष्यतुष्ट्यै ॥
² See Sudhâkara Dvivedî, Gaṇaka-taraṅgiṇî (or Lives of Hindu Astro-
nomers), Banaras (1933), pp. 22, 24.
³ The date of this work is not known, but it was written earlier than
the Jâtaka-paddhati of Śrîpati (1039 A.D.).
⁴ See S. B. Dîkṣita, l. c., p. 312.
⁵ See Vidyâmâdhavîyam by Vidyâmâdhava. Edited with the commen-
tary, entitled Muhûrta dîpikâ, of Viṣṇu Śarmâ by R. Shamasastry, Mysore
(1923, 1925, 1926).
⁶ See N. C. Jaina, 'Śrîdharâcârya', Jaina-Siddhânta-Bhâskara, Vol. 14,
No. 1, pp. 31-42. Also see S. Śrîkantha Śâstrî, 'The date of Śrîdharâcârya',
The Jaina Antiquary, Vol. XIII, No. 2, pp. 12-17, for the criticism of N. C.
Jaina's views.

INTRODUCTION xv
(iii) Jyotirjñāna-vidhi, or Śrīkaraṇa, written in 799 A.D.
But we have shown below that Śrīdharācārya was a
Śaivite Hindu, not a Jaina, and that he lived sometime bet-
ween Mahāvīra (850 A.D.) and Āryabhaṭa II (c 950 A.D.)
So he cannot be taken to be the author of any of these works.
3·1. The Patiganita. The Pāṭīgaṇita aims at provid
ing a complete course of arithmetic and mensuration to
cater the needs of both students and businessmen. It is
divided as usual into two main sections, viz., logistics and
determinations. The author gives in the beginning of the
work the names of the twenty-nine logistics and the nine
determinations which he proposes to deal with in this work.
These are as follows :
Logistics : (1) saṅkalita (addition), (2) vyvakalita (subtrac-
tion, (3) pratyutpanna (multiplication), (4) bhāgahāra (division),
(5) varga (square), (6) varga-mūla (square root), (7) ghana
(cube), (8) ghana-mūla (cube root), (9-16) the same operations
for fractions, (17-22) kalāsavarṇa (reduction of fractions of six
varieties), (23) trairāśika (rule of three), (24) vyasta-trairāśika
(inverse rule of three), (25) pañca-rāśika (rule of five), (26)
sapta-rāśika (rule of seven), (27) nava-rāśika (rule of nine), (28)
bhāṇḍa-pratibhāṇḍa (barter of commodities), and (29) jīva-vikraya
(sale of living beings).
Determinations : (1) miśraka (mixtures), (2) śreḍhī (series),
(3) kṣetra (plane figures), (4) khāta (excavations), (5) citi (piles
of bricks), (6) krākaca (sawn pieces of timber), (7) rāśi (heaps
or mounds of grain), (8) chāyā (shadow), and (9) śūnya-tatva
(the mathematics of zero).
Of the above twenty-nine logistics, the two, viz., bhāga-
mātā and jīva-vikraya were not treated by Brahmagupta. The
rule of eleven which was included in the list of logistics by
Brahmagupta and on which an example was set by Pṛthūda-
kasvāmī, has been omitted by Śrīdharācārya, probably be-

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.