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

CONTENTS
सुवर्णगणितम् ६३
हेमैक्यवर्णः ६३—पक्ववर्णः ६४—पक्वसुवर्णः ६६—नष्टवर्णः ६७—
अज्ञातसुवर्णः ६८—वर्णमालिका ६९—हेमपरिमाणविभागः गुड़िका-
वर्णः ७१
प्रक्षेपगणितम् ७३
प्रक्षेपफलं ७३—विजातीयविषयः ७५ ।
विविधविषयाः ७६
क्रयविक्रयौ ७६—शेषार्धच्छेदविशेषः ७८—निर्दिष्टमूल्येन निर्दिष्ट-
सङ्ख्यान्तर्गतजीवपण्यक्रयः ७९—मन्दशीघ्रगतिमेलापकालः ८४—
गमागमसङ्गमकालः ८६—मार्गप्रमाणं ८७—अन्यगतिज्ञानं ८७—
भाटकः ८७—व्यावृत्तपुरुषदायग्रहणविभागः ८८ - वापीपूरणकालः
९४—अन्तर्भाटकः ९५—मार्गविभागः ९६—रसभेदाः ९७—रस-
प्रस्तारः ९९—स्तम्भ-शेषोद्देशकौ १०१—विशेषोद्देशकः १०२—
मूलादिशेषोद्देशकः १०३—भागमूलाग्रोद्देशकः १०४—उभयाग्रमूल-
शेषोद्देशकः १०५—विपरीतोद्देशकः १०६ ।
श्रेढीव्यवहारः १०७
श्रेढीस्वरूपं १०७—श्रेढीफलं ११०—आदिः ११८—चयः ११९—
गच्छः १२०—आद्युत्तरपृथक्करणं १२५--सविकलसङ्कलितं १२६
—सविकलपदादिः १२८—सविकलपदवृद्धिः १२९—सविकलपदं
१३०—गुणोत्तरसङ्कलितं १३४—आद्यन्तधनपद-सङ्कलितं १३६—
तुल्यगतिकालः १३६—मेलकद्वितयं १४१--द्यूतजयपराजयौ १४५—
सङ्कलितकृतिघनैक्यं १४८—वर्गसङ्कलितं १५०—घनसङ्कलितं १५०
—सङ्कलितसङ्कलितं १५१—सङ्कलितकृतिघनसङ्कलितैक्यं १५१—
इष्टाद्युत्तरवर्गसङ्कलितं १५२—इष्टाद्युत्तरसङ्कलितसङ्कलितं १५३
—इष्टाद्युत्तरघनयोगः १५३ ।
क्षेत्रव्यवहारः १५५
क्षेत्रसंभवासंभवौ १५६—क्षेत्रवस्तुलाभालाभौ १५६—क्षेत्रविभागः
१५६—स्थूलसूक्ष्मफले १५६—समलम्बचतुरश्रत्र्यश्रक्षेत्रफले १६१—
गजदन्तादिक्षेत्रफलानयनोपयोगिक्षेत्रविभागः १७१—श्रवणालम्बज्ञाना-
भावे त्र्यश्रचतुरश्रक्षेत्रफले १७५—आसन्नमूलम् १७५ ।
शब्दानुक्रमणिका १७७

CONTENTS
English Translation
Page
Homage and Introduction 1
Contents 1
Definitions 2–4
Names of notational places 2 —Table of money-mea-
sures 3 —Table of weights 3 —Table of measures of
capacity 4 —Table of linear measures 4 —Table of
time-measures 4.
Logistics ( parikarma ) 5–30
Addition 5 —Subtraction 6 —Multiplication 7 —
Division 8 —Squaring 8 —Square root 9 —Cubing
11 —Cube root 12 —Operations for fractions
15 —Reductions of fractions 17 —Rule of three
22 —Inverse rule of three 25 —Rules of five, seven
and nine 25 —Barter of commodities 28 —Sale of
living beings 29.
Determinations pertaining to mixtures of things 31–65
Simple interest 31 —Alligation of pieces of gold
36—Partnership 42 —Purchase and sale 43 —
Meeting of two travellers 52 —Wages and pay-
ments 53 —The cistern problem 55 —Wages paid
from the commodity 56 —Combination of savours
58 —Certain special types of problems 59.
Determinations pertaining to series 66–85
Series in arithmetic progression 66 —Series in
geometric progression 75 —Miscellaneous problems
on series in arithmetic progression 77 —Series of
squares, cubes and successive sums of natural num-
bers 82 —Series of squares and cubes etc. of the
terms of an arithmetic series 84.
Determinations pertaining to plain figures 86–91
Introduction 86 —Area of the quadrilateral with
equal altitudes and of the triangle 88 —Area of
the quadrilateral with unequal altitudes 90.
Answers to Examples 93–96
Indological Truths

ABBREVIATIONS
 Āryabhaṭīya
BM Bakhshālī Manuscript
BBî (ASS) Bhāskara II's Bījagaṇita (Ānandāśrama Sanskrit Series)
BrSpSi Brāhma-sphuṭa-siddhānta
GK Gaṇita-kaumudī
GSS Gaṇita-sāra-saṅgraha
GT Gaṇita-tilaka
L(ASS) Līlāvatī (Ānandāśrama Sanskrit Series)
MSi Mahā-siddhānta
PG Pāṭīgaṇita
SiŚe Siddhānta-śekhara
Triś Triśatikā

INTRODUCTION

1. The present work contains the Sanskrit text of
   Śrîdharâcârya's Pâṭīgaṇita along with an ancient Sanskrit
   commentary and an English translation with relevant notes
   and comments. Śrîdharâcârya's Pâṭīgaṇita deals with
   arithmetic and mensuration and is the author's bigger work
   on the subject. The existence of this work was known to
   earlier scholars working in the field of Hindu Mathematics,
   such as Sudhâkara Dvivedî and S.B. Dîkṣita, from the author's
   own allusion to this work in the opening stanza of his smaller
   work on the subject, which was edited by Sudhâkâra Dvivedî
   in 1899 A.D. under the name of Triśatikā, but it was not
   available to them either in manuscript or in print. Although
   a manuscript of this work existed in the Raghunâtha Temple
   Library at Jammu, it remained unnoticed by the early
   scholars. The late Dr A. N. Singh was the first to discover
   in this manuscript the text of Śrîdharâcârya's bigger work on
   arithmetic and mensuration. While consulting the Catalogue
   of the above-mentioned Library, he came across a manuscript
   which was noticed as follows :¹
   Manuscript Number 3074
   Name of Work Pâṭīgaṇita-ṭīkâ
   Name of Author Śrîdharâcârya
   Number of leaves 157
   Lines per page 9
   Syllables per line 44
   Time of writing the manuscript —
   Remarks—Incomplete. Modern Kashmir script. Name
   of commentator unknown.
   Thinking that it might be a manuscript of Śrîdharâcârya's
   bigger work on arithmetic and mensuration, he ordered a
   ¹ See Catalogue of the Sanskrit Manuscripts in the Raghunâtha Temple
   Library of His Highness the Maharaja of Jammu and Kashmir, prepared by
   M.A. Stein, Bombay (1894), p. 165.

ii INTRODUCTION
copy of that manuscript. When the copy was received and
examined, his conjecture came out to be true. The occur-
rence in the manuscript of the verses which were ascribed
by the later writers to Śrīdharâcârya’s Pâtīgaṇita proved
beyond doubt that it was the manuscript of the very same
work. This is how the manuscript of the present work was
discovered.
Soon after the discovery of this important manuscript,
I joined Dr. Singh in his researches on Hindu Mathematics.
Both of us felt the necessity of publishing this manuscript, but
as the manuscript was incomplete and suffered from errors
and inaccuracies, it was considered advisable to wait for some
time for the discovery of another complete and better manu-
script. We waited, but inspite of our best efforts in this
direction another manuscript could not be traced. We there-
fore decided to edit the available manuscript, however incom-
plete and defective it was. Due to other work in hand,
however, the task of editing this manuscript could not
be undertaken during the lifetime of Dr Singh. Although
the text of the manuscript was already studied by the present
editor during 1948-49 while helping Dr Sabal Singh of
Balwant Rajput College, Agra, in writing his thesis on
Śrīdharâcârya, the regular work of editing the manuscript
was started only in July, 1957.
We have seen above that the available manuscript is in-
complete. Let us now describe it more fully. It starts with
an invocation to God Gaṇeśa, Lord of Success, by the commen-
tator, and runs up to the twenty-third verse of the Kṣetra-
vyavahâra (‘Determinations pertaining to plain figures’). The
available portion of the work comprises the following topics :
(1) Definitions, (2) Logistics (parikarma), (3) Deter-
minations pertaining to mixtures of things (miśraka-vyavahâra),
(4) Determinations pertaining to series (śreḍhī-vyavahâra), and
(5) Determinations pertaining to plain figures (kṣetra-vyavahâra),
the last topic being incomplete.

INTRODUCTION iii
The following topics, though mentioned by the author
in the list of contents, are totally missing from the manuscript :
(1) Determinations pertaining to excavations (khâta-
vyavahâra), (2) Determinations pertaining to piles of bricks
(citi-vyavahâra), (3) Determinations pertaining to sawn pieces
of timber (krâkaca-vyavahâra), (4) Determinations pertaining
to heaps of grain (râśi-vyavahâra), (5) Determinations pertain-
ing to shadow (châyâ-vyavahâra), and (6) The mathematics of
zero (śûnya-tatva).
The available text of the Pâṭīgaṇita comprises altogether
251 verses, of which 118 contain definitions and rules and
133 contain examples.
The manuscript is fairly complete in so far as it goes.
A serious lacuna, however, occurs after verses 18-20 embodying
rules for multiplication, which extends over the entire com-
mentary to those three verses and also over the following exam-
ple and part of the commentary to that example. There was,
however, little difficulty in substituting the missing example
from the author's smaller work on the subject. The surviving
commentary to the missing example was sufficient to indicate
which example it was. Thus inspite of the above-mentioned
lacuna little damage was caused to the text of the Pâṭīgaṇita,
only a little portion of the commentary was lost.
The editing of an ancient Sanskrit work on the basis of
of a single manuscript, especially when it is full of errors and
inaccuracies, is indeed a difficult task. In editing the above
manuscript we have followed the principle of least interference
and have aimed at removing the defects of the manuscript
by making as few alterations as possible. With this end in
view we had to read the manuscript several times to find the
correct meaning of the text so as to be able to ascertain the
nearest substitutes for the incorrect words or phrases occurring
in the manuscript. As far as the numerical figures were con-
cerned there was not much difficulty in rectifying them, as
they could be verified mathematically. In the case of non-

iv INTRODUCTION
mathematical words or phrases, where we had to face some
difficulty, we were guided by the criterion of relevance and
appropriateness. Where alterations and modifications have
been made, the original readings have been given in the foot-
notes, so that the interested reader may himself decide the
appropriateness of the alterations and modifications made by
us. Whenever missing or explanatory words have been inser-
ted, they have been enclosed within small brackets ( ).
Portions of the commentary which appeared to us irrelevant
have been enclosed within square brackets [ ] .
The verses of the text were not numbered in the manu-
script. But for the facility of reference we felt the desirability
of numbering them. Here also we had to face some difficulty.
We found that verses containing examples were in many cases
lying between parts of verses containing rules, and further
that the two types of verses were not in the same metre, so
that it was impossible to provide a continuous numbering to
the whole text. We had therefore to number the verses con-
taining examples separately from these containing definitions
and rules, as Sudhâkara Dvivedî had done in numbering the
verses of the Triśatikâ. To distinguish the two types of
verses and their numbering we have got them printed in
different points.
For our English-knowing readers, who are not versed in
Sanskrit, we have rendered the whole text of the Pâṭīgaṇita
into English. Here we have aimed at giving as far as possible
a literal version of the text. Technical terms which have
their English equivalents have been translated into English ;
others have been kept as they are and have been explained.
The portions of the English translation enclosed within
brackets do not occur in the text and have been given in the
translation to make it understandable, and are at places expla-
natory. Without these portions the translation at places would
appear meaningless to the reader who cannot consult the
original for lack of knowledge of Sanskrit. We have tried our

best to keep the spirit of the original and have as far as possible
not altered the sequence in the translation.
The translation to each rule is preceded by a sentence
or two giving in brief the contents of the rule and is followed,
where necessary, by relevant notes and comments. Parallel
rules and examples found in other available works on Hindu
mathematics have been indicated in the footnotes. Headings
and sub-headings have been provided to facilitate consulta-
tion.
2. PATIGANITA. Pâtîgaṇita is the name given to that
branch of Hindu Mathematics which deals with arithmetic
and mensuration. It is believed that this subject attained an
independent status sometime before the beginning of the
Christian era, when arithmetic became a separate subject and
geometry, which formerly belonged to a separate group of
Sciences, viz., the Kalpasûtra, came to be incorporated with it.¹
The earliest work exclusively dealing with this subject which
has come down to us is the fragmentary manuscript known as
the Bakhshâlî Manuscript² (text composed about 200 A.D.), which
was discovered in 1881 A.D., at Bakhshâlî, a village near the
city of Peshawar in the north-west of India, in course of ex-
cavation by a farmer. This work shows beyond all doubt that
by the third century A.D., when its text was composed, the
subject of Pâtîgaṇita was already in a state of maturity.
There are references to a number of other works on the
subject which were in use in the fifth and sixth centuries in
India. Bhâskara I (629 A.D.) refers to the works of Maskarî
Pûraṇa, Mudgala, Patana, and others, which were exclusively
devoted to the subject of Pâtîgaṇita. In the Âryabhaṭîya of
Âryabhaṭa I (b. 476 A.D.), written about the end of the
fifth century A.D., there is a chapter which deals with
¹ See B. Datta, 'The Scope and Development of the Hindu Gaṇita,'
Indian Historical Quarterly, Vol. V, No. 3, Sept. 1929, pp. 478-512.
² Bakhshâlî Manuscript—A Study in Mediaeval Mathematics. Edited
by G. R. Kaye. Parts I and II, Calcutta (1927). Part III, Delhi (1933).

vi INTRODUCTION
mathematics and is called Gaṇita-pâda, but this is too brief
and does not give a clear idea of the state of mathematical
knowledge in India in those times. According to Bhâskara I,
the most competent scholiast of Âryabhaṭa I, the subject-
matter discussed by Âryabhaṭa I in his Gaṇita-pâda hardly
deserves the name mathematics (gaṇita). He has preferred to call
it 'a bit of mathematics.' He writes : "In the Gaṇita-pâda
the Âcârya (Âryabhaṭa) has dealt with the subject of mathe-
matics (gaṇita) by indications only, whereas in the Kâlakriyâ-
pâda and Gola-pâda¹ he has discussed 'reckoning with time'
(kâlakriyâ) and 'spherical astronomy' (gola) in detail. So by
the word gaṇita (used by Âryabhaṭa) one must understand 'a
bit of mathematics.' Otherwise, the subject of mathematics is
vast. There are eight vyavahâras (determinations), miśraka
(mixtures), śreḍhî (series), kṣetra (plane figures), khâta (excava-
tions), citi (piles of bricks), krâkacika (saw problems) and châyâ
(shadow). The miśraka is that which involves the mixture of
several things. The śreḍhî is that which has a beginning (i.e.,
first term) and an increase (i.e., common difference). The
kṣetra tells us how to calculate the area of a figure having
several angles. The khâta enable us to know the volumes of
excavations. The citi tells us the measure of a pile in terms
of bricks. The krâkacika : The krâkaca (saw) is a tool which
saws timber ; that which relates to the sawing of timber, i.e.,
that which tells the measure of the timber sawn, is called
krâkacika (vyavahâra). The râśi tells us the amount of a heap
of grain, etc. The châyâ tells us the time from the shadow of
a gnomon, etc. Of the vyavahâra-gaṇita (practical or commer-
cial mathematics, i.e., pâṭîgaṇita), which is thus of eight
varieties, there are four bîjas, viz., first, second, third, and
fourth, i.e., yâvattâvat ('theory of the simple equation'), vargâvarga
('theory of the quadratic equation'), ghanâghana ('theory of the
cubic equation'), and viṣama ('theory of the equations with
several unknowns'). Rules and examples pertaining to each
one of these have been compiled (in independent works) by
¹ These are the names of chapters three and four of the Âryabhaṭîya.

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.