Elementary Operations, Fundamental operations · 2014-04-14 · Elementary Operations, Fundamental operations: On the parikarman's of Sanskrit mathematical texts VIIth-XIIth century

Elementary Operations, Fundamental operations: On the parikarman's of Sanskrit mathematical texts

VIIth-XIIth century.

14/11/2011

Agathe Keller

Monday, November 14, 2011

Bhāskarācārya (b.1114), Siddhānta Śiromāṇiincluding Līlāvatī and Bījagaṇita

H. T. Colebrooke. Algebra, with Arithmetic and mensuration. J. Murray, London, 1817.F. Patte. Le Siddhāntaśiromāni : l’oeuvre mathématique et astronomique de Bhāskarācarya =

Siddhāntaśiromāniḥ : Śri-Bhāskarācarya-viracitaḥ. Droz, 2004.T. Hayashi. Bījagaṇita of Bhāskara. SCIAMVS (10): 2009.

Śrīdharācārya (ca.900), Pāṭīgaṇitaand anonymous undated commentary

K. S. Shukla. Pāgaṇita of Śrīdharācarya. Lucknow University, Lucknow, 1959.

Brahmagupta (629) BrahmasphuṭṭasidhāntaPṛthūdakṣvamin (ca.980) Vāsanābhāṣya

H. T. Colebrooke. Algebra, with Arithmetic and mensuration. J. Murray, London, 1817.M. S. Dvivedin. Brāhmasphuṭasiddhānta and Dhyānagrahopadeṣādhyāya by Brahmagupta edited with his own

commentary The paṇḍit, XXIV:454, 1902.S. Ikeyama. Brāhmasphuasiddhānta (ch. 21) of Brahmagupta with Commentary of Pṛthudaka, critically ed. with

Eng. tr. and Notes, volume 38. Indian Journal of History of Science, New Delhi, India, 2003.


CONTENTS.

Page.

DiSSERTATIOJf • i

NOTES AND ILLUSTRATIONS.

A. Scholiasts of Bha'scara xxv

B. Astronomy of Brahmegupta . . xxviii

C. Brahma-sidd'hdnta, Title of his Astronomy xxx

D. Verification of his Text xxxi

E. Chronology of Astronomical Authorities, according to Astrono-

mers of Ujjayani xxxiii

F. Age of Brahmegupta, from astronomical data xxxv

G. Aryabhatta's Doctrine xxxvii

H. (Reference from p. ix. 1.21.) Scautlness of Additions by later

Writers on Aljgfpbra xl

I. Age of AnrABHATTA xU

K. Writings and Age of Vara'ha-mihira xlv

L. Introduction and Progress of Algebra among the Italians ... li

M. Arithmetics of Diophantus Ixi

N. Progress and Proficiency of the Arabians in Algebra Ixiv

O. Communication of the Hindus with Western Nations on Astro-

logy and Astronomy Ixxviii

BHASCARA.

ARITHMETIC (Lildvatl)

Chapter I. Introduction. Axioms. Weights and Measures . . 1

Chapter II. Sect. I. Invocation. Numeration 4

Sect. II. Eight Operations of Arithmetic : Addition, &c. 5

Sect. III. Fractions 13

Sect. IV. Cipher 19

I

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

CONTENTS.

III. Afiscellaneous.

Sect. I. Inversion 21

Sect. II. Supposition 23

Sect. III. Concurrence . ; 26

Sect. IV. Problem concerning Squares ... 27

Sect. V. Assimilation 29

Sect. VI. Rule of Proportion 33

IV. Mixture.

Sect. I. Interest 39

Sect. II. Fractions 42

Sect. III. Purchase and Sale 43

Sect. IV. A Problem 45

Sect. V. Alligation 46

Sect. VI. Permutation and Combination ... 49

V. Progression.

Sect. I. Arithmetical 51

Sect. II. Geometrical 55

VI. Plane FSgure 58

VII. Excavations and Content of Solids 97

VIII. Stacks 100

IX. Saw 101

X. Mound of Grain i03

XI. Shadow of a Gnomon 106

XII. Pulverizer fCM^/aca) 112

XIII. Combination 123

ALGEBRA (Vija-gatiita.)

Chapter I. Sect. I. Invocation, &c 129

Sect. II. Algorithm ofNegative and Affirmative

Quantities 133

Sect. III. of Cipher 136

Sect. IV. of Unknown Quantity . . ] 39

Sect. V. of Surds 145

Chapter II. Pulverizer 156

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

CONTENTS.








IV. Mixture.







V. Progression.



VI. Plane FSgure 58


VIII. Stacks 100

IX. Saw 101








Quantities 133




Chapter II. Pulverizer 156CONTENTS.

Chapter III. Affected Square. Sect. I. 170

Sect. II. Cyclic Method '.. 175

Sect. III. Miscellaneous 179

Chapter IV. Simple liquation 185

Chapter V. Quadratic, &c. Equations 207

Chapter VI. Multiliteral Equations 227

Chapter VII. Varieties of Quadratics 245

Chapter VIII. Equation involving a Factum of Unknown Quantities 268

Chapter IX. Conclusion 275

BRAHMEGUPTA.

CHAPTER XII. ARITHMETIC (Gariita.)

Sect. I. Algorithm 277

Sect. II. Mixture 287

Sect. III. Progression 29()

Sect. IV. Plane Figure 305

Sect. V. Excavations 312

Sect. VI. Stacks 314

Sect. VII. Saw 315

&c^ F7//. Mounds of Grain 316

Sect. IX. Measure by Shadow 317

Sect. X. Supplement 319

CHAPTER XVIII. ALGEBRA (CuHaca.)

Sect. I. Pulverizer 325

Sect. II. Algorithm 339

Sect. III. Simple Equation 344

Sect. IV. Quadratic Equation 346

Sect. V. Equation of several unknown . . . 348

Sect. VI. Equation involving a factum ... 361

Sect. VII. Square affected by coefficient . . . 363

Sect. VIII. Problems 373


Structure de la Lilavati Structure du Bijaganita

ghanamūla

8 operations 8 practices

+

-

x

÷

a

!

!

2

a

a3

3

a

+..

+...

+

!

◎△ !

!

"

#

6 methods 4 equations

+

-

x

÷

a

!

2

a

x

!

x

x

y

2

saṅkalita

parikarman

vyavakalita

pratyutpanna

bhāgahāra

varga

vargamūla

ghana

ghanamūla

vyavahāra


BG.2.1 Since the visible [mathematics] (i.e., mathematics with known numbers) told before [by me in the L] has the invisible [mathematics] as its seed, and since, without reasoning of the

invisible [mathematics], problems can hardly be understood (i.e., solved) [even by intelligent persons and] not at all by less-intelligent persons, I speak about the operations with seeds.

BG 2 pūrvaṃ proktaṃ vyaktaṃ avyaktabījaṃ prāyaḥ praśnāḥno vināvyaktayuktyā/

jñātum śakyāḥ mandadhībhir nitāntam yasmāt tasmāt vacmi bījakriyāṃ ca//


6 rules

Positives

and

negatives

8 operations

Whole

numbers

Fractions

Zero

Zero

Undeter-

mined

surds

+ +

- -

x x÷ ÷

a a a! !

!

2 23

3

rūpa

bhinna

śunya

ṛṇa dhana

śunya

avyakta

karaṇī


CONTENTS.

Page.

DiSSERTATIOJf • i

NOTES AND ILLUSTRATIONS.

A. Scholiasts of Bha'scara xxv

B. Astronomy of Brahmegupta . . xxviii

C. Brahma-sidd'hdnta, Title of his Astronomy xxx

D. Verification of his Text xxxi

E. Chronology of Astronomical Authorities, according to Astrono-

mers of Ujjayani xxxiii

F. Age of Brahmegupta, from astronomical data xxxv

G. Aryabhatta's Doctrine xxxvii

H. (Reference from p. ix. 1.21.) Scautlness of Additions by later

Writers on Aljgfpbra xl

I. Age of AnrABHATTA xU

K. Writings and Age of Vara'ha-mihira xlv

L. Introduction and Progress of Algebra among the Italians ... li

M. Arithmetics of Diophantus Ixi

N. Progress and Proficiency of the Arabians in Algebra Ixiv

O. Communication of the Hindus with Western Nations on Astro-

logy and Astronomy Ixxviii

BHASCARA.

ARITHMETIC (Lildvatl)

Chapter I. Introduction. Axioms. Weights and Measures . . 1

Chapter II. Sect. I. Invocation. Numeration 4

Sect. II. Eight Operations of Arithmetic : Addition, &c. 5

Sect. III. Fractions 13

Sect. IV. Cipher 19

I

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

CONTENTS.








IV. Mixture.







V. Progression.



VI. Plane FSgure 58


VIII. Stacks 100

IX. Saw 101








Quantities 133




Chapter II. Pulverizer 156

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

CONTENTS.








IV. Mixture.







V. Progression.



VI. Plane FSgure 58


VIII. Stacks 100

IX. Saw 101








Quantities 133




Chapter II. Pulverizer 156CONTENTS.

Chapter III. Affected Square. Sect. I. 170

Sect. II. Cyclic Method '.. 175

Sect. III. Miscellaneous 179

Chapter IV. Simple liquation 185

Chapter V. Quadratic, &c. Equations 207

Chapter VI. Multiliteral Equations 227

Chapter VII. Varieties of Quadratics 245

Chapter VIII. Equation involving a Factum of Unknown Quantities 268

Chapter IX. Conclusion 275

BRAHMEGUPTA.

CHAPTER XII. ARITHMETIC (Gariita.)

Sect. I. Algorithm 277

Sect. II. Mixture 287

Sect. III. Progression 29()

Sect. IV. Plane Figure 305

Sect. V. Excavations 312

Sect. VI. Stacks 314

Sect. VII. Saw 315

&c^ F7//. Mounds of Grain 316

Sect. IX. Measure by Shadow 317

Sect. X. Supplement 319

CHAPTER XVIII. ALGEBRA (CuHaca.)

Sect. I. Pulverizer 325

Sect. II. Algorithm 339

Sect. III. Simple Equation 344

Sect. IV. Quadratic Equation 346

Sect. V. Equation of several unknown . . . 348

Sect. VI. Equation involving a factum ... 361

Sect. VII. Square affected by coefficient . . . 363

Sect. VIII. Problems 373


8 operations

+

-

x ÷

a a!

!

2 3

3

8 practices

+..+...+!!!

△ ◎" #mixtures

Proportions

Concurence

Suppostion

Inversion

Combinatorics

pulverizer

Barter and Exchange

saṃkramaṇa

kuṭṭaka

trairāśikādi


PULVERIZER. 157

measure, they are termed reduced quantities. Divide mutually the reduced

dividend and divisor, until unity be the remainder in the dividend. Place

the quotients one under the other; and the additive quantity beneath them,

and cipher at the bottom. By the penult multiply the number next above

it, and add the lowest term. Then reject the last and repeat the operation

until a pair of numbers be left. The uppermost of these being abraded by

the reduced dividend, the remainder is the quotient. The other [or lower-

most] being in like manner abraded by the reduced divisor, the remainder is

the multiplier.'

could not measure the remaining portion or residue, if it were greater than it. When therefore the

greater number, divided by the lesi, yields a residue, the greatest common measure, in such case,

is equal to ihe remainder, provided this be a measure of the less. If again the less number, divided

by the remainder, yield a residue, the common measure cannot exceed this residue ; for the same

reason. Therefore no number, though less than the first remainder, can be a common measure, if

it exceed the second remainder: and the greatest common measure is equal to the second remain-

der, provided it measure the first; for then of course it measures the multiple of it, which is the

other portion of the second number. So, if there be a third remainder, the greatest common mea-

sure is either equal to it, if it measure the second ; or is less. Hence the rule, to divide the greater

number by the less, and the less by the remainder, and each residue by the remainder following,

until a residue be found, which exactly measures the preceding one; such last remainder is the

common measure. (§ 54). CRfsHN.

' The substance of Cri'shn'a's demonstration is as follows: When the dividend, taken into the

multiplier, is exactly measured by the divisor, the additive must either be null or a multiple of the

divisor. (See § 63). If the dividend be such, that, being multiplied by the multiplicator and di-

vided by the divisor, it yields a residue, the additive, if negative, must be equal to that remainder;

(and then the subtractive quantity balances the residue;) or, if affirmative, it must be equal to the

difference between the divisor and residue;(and so the addition of that quantity completes the

amount of the divisor;) or else it must be equal to the residue, or its complement, with the divisor

or a multiple of the divisor added. Let the dividend be considered as composed of two portions

or terms: 1st, a multiple of the divisor; 2d, the overplus or residue. The first multiplied by the

multiplier (whatever this be), is of course measured by the divisor. As to the second, or overplus

and remainder, the additive being negative, both disappear when the multiplier is quotient of the

additive divided by the remainder, (the additive being a multiplier of the residue.) But, if the

additive be not a multiple of the remainder, should unity be the residue at the first step of the re-

ciprocal division, the multiplier will be equal to the additive, if this be negative, or to its comple-

ment to the divisor, if it be positive; and the corresponding quotient will be equal to the quotient

of the dividend by the divisor multiplied by the multiplicator, if the additive be negative; or be

equal to the same with addition of unity, if it be affirmative: and, generally, when reciprocal divi-

sion has reached its last step exhibiting a remainder of one, the multiplier, answering to the pre-

ceding residue, become the divisor, as serving for that next before it become dividend, is equal to

CHAPTER II.

PULVERIZER.'

53—64. 'Rule: In the first place, as preparatory to the investigation of

the pulverizer, the dividend, divisor, and atlditive quantity are, if practicable,

to be reduced by some number.' If the number, by which the dividend and

divisor are both measured, do not also measure the additive quantity, the

question is an ill put [or impossible] one.*

54

—

55—56. The last remainder, when the dividend and divisor are mu-

tually divided, is their common measure.* Being divided by that common

* This is nearly word for word the same with a chapter in the Lildtati on the same subject.

(Li7. Ch. 12.) See there, explanations of the terras.

The method here taught is applicable chiefly to the solution of indeterminate problems that pro-

duce equations involving more than one unknown quantity. See ch. 6.

* Ten stanzas and two halves.

* If the dividend and divisor admit a common measure, they must be first reduced by it to their

least terras; else unity will not be the residue of reciprocal division; but the common measure

will; (or, going a step further, nought.)

—

Ga'n. on Lil. Crishn. on Vy.

* If the dividend and divisor have a common measure, the additive also must admit it; and the

three terms be correspondently reduced: for the additive, nnkss it be [nought or else] a multiple

of the divisor, must, if negative, equal the residue of a division of the dividend taken into the mul-

tiplier by the divisor; and, if affirmative, must equal the complement of that residue to the divisor.

Now, if dividend and divisor be reducible to less terras, the residue of division of the reduced terms,

multiplied by the common measure, is equal to the residue of division of the unreduced terms.

Therefore the additive, whether equal to the residue, or to its complement, must be divisible by the

common measure. Crishn.

* The common measure may equal, but cannot exceed, the least of the two numbers:for it

must divide it. If it be less, the greater may be considered as consisting of two terras, one the

quotient taken into the divisor, the other the residue. The common measure cannot exceed that

residue; for, as it measures the divisor, it must of course measure the multiple of the divisor, and

Pulveriser


Datta Singh p. 128:

“The eight “fundamental” operations of Hindu gaṇita are (1) addition, (2) subtraction, (3)

mutliplication, (4) division, (5) square, (6) square-root, (7) cube, (8) cube-root.

Most of these elementary processes have not been mentioned in the Siddhânta works”















6 rules

Positives

and

negatives

8 operations

Whole

numbers

Fractions

Zero

Zero

Undeter-

mined

surds

+ +

- -

x x÷ ÷

a a a! !

!

2 23

3

rūpa

bhinna

śunya

ṛṇa dhana

śunya

avyakta

karaṇī

4 fold class


bhāgajāti part class

a1

b1

a2

b2+

LV 30 anyonyahārābhihatau harāṃśau rāśyor samachedavidhānam evaṃ\\mithas harābhyām apavartitābhyām yadvā harāṃśau sudhiyā atra guṇyau//\\

The numerator and denominator being multiplied reciprocally by the denominators of the two quantities, they are thus reduced to the same denominators. Or both numerator and denominator may be multiplied by the intelligent calculator into the reciprocal denominators abridged by a common measure.

a1

b1

a2

b2

+b2 b1

31

15+ 15

53+ 13


prabhāgajāti different part class a1

b1x a2

b2

LV032 lavāllavaghnāś ca harāḥ haraghnāḥ bhāgaprabhāgeṣu savarṇanam syāt\\The numerators multiplied by the numerators, and the denominators by the denominators will be same-

coloured when [they are] different parts .

a1

b1

a2

b2

a1

b1

a2

b2x

11

12x

128012

3x 3

4x 116

x 14x


(sva)bhāgānubandhaṃjāti one’s own part additive class

a2

b2

a1

b1

kālasaṃvarṇa samecoloured portion

a1

b1

n

LV034 chedaghnarūpeṣu lavāḥ dhanaṛṇam ekasya bhāgās adhikaunakāś ced//svāṃśādhikaunas khalu yatra tatra bhāgānubandhe ca lavāpavāhe/talasthahāreṇa haram nihanyāt svāṃśādhikaunena tu tena bhāgān//The integer being multiplied by the denominator, the numerator is made positive or negative, provided parts of an unit be added or be subtractive. But, if indeed the quantity be increased or diminished by a part of itself, then, in the addition and subtraction of fractions, multiply the denominator by the denominator standing underneath, and the numerator by the same augmented or lessened by it own numerator.

a1(b2 +a2)b1 b2

a1/b1 + (a1/b1 x a2/b2) n+a1/b1

b1 n+a1

b1


svabhāgāpavahojāti one’s own subtractive part class

bhāgāmātajāti mother part class

-a2

a1

b2

b1-a1

b1

n

LV034 chedaghnarūpeṣu lavāḥ dhanaṛṇam ekasya bhāgās adhikaunakāś ced//svāṃśādhikaunas khalu yatra tatra bhāgānubandhe ca lavāpavāhe/talasthahāreṇa haram nihanyāt svāṃśādhikaunena tu tena bhāgān//The integer being multiplied by the denominator, the numerator is made positive or negative, provided parts of an unit be added or be subtractive. But, if indeed the quantity be increased or diminished by a part of itself, then, in the addition and subtraction of fractions, multiply the denominator by the denominator standing underneath, and the numerator by the same augmented or lessened by it own numerator.

a1 (b2 -a2)b1 b2

b1 n-a1

b1

a1/b1 - (a1/b1 x a2/b2) n+a1/b1


bhinnasaṅkalita addition of fractionsLV037. yogas antaram tulyaharāṃśakānām kalpyas haras rūpam ahārarāśeḥ//The sum or the difference of fractions having the a common denominator, is [taken]. Unity is put denominator of a quantity which has no divisor.












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Elementary Operations, Fundamental operations · 2014-04-14 · Elementary Operations, Fundamental operations: On the parikarman's of Sanskrit mathematical texts VIIth-XIIth century