Algebra, with Arithmetic and Mensuration

by Brahmagupta

ALGEBRA,

WITH

ARITHMETIC AND MENSURATION,

FROM THE

SANSCRIT

OF

BRAHMEGUPTA AND BHASCARA.

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TRANSLATED BY

HENRY THOMAS COLEBROOKE, Esq.

F.R.S.; M. LINN. AND GEOL. SOC. AND R. INST. LONDON; AS. SOC. BENGAL

AC. SC. MUNICH.

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LONDON:

JOHN MURRAY, ALBEMARLE STREET

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1817.

ALGEBRA,

WITH

ARITHMETIC AND MENSURATION,

FROM THE

SANSCRIT.

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London: Printed by C. Roworth,

Bell-yard, Temple-bar.

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CONTENTS.

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			Page.
Dissertation			i
NOTES AND ILLUSTRATIONS.
A. Scholiasts of Bháscara			xxv
B. Astronomy of Brahmegupta			xxviii
C. Brahma-sidd'hánta, Title of his Astronomy			xxx
D. Verification of his Text			xxxi
E. Chronology of Astronomical Authorities, according to Astronomers of Ujjayaní			xxxiii
F. Age of Brahmegupta, from astronomical data			xxxv
G. Áryabhat́t́a's Doctrine			xxxvii
H. (Reference from p. ix. 1.21.) Scantiness of Additions by later Writers on Algebra			xl
I. Age of Áryabhat́t́a			xli
K. Writings and Age of Varáha-mihira			xlv
L. Introduction and Progress of Algebra among the Italians			li
M. Arithmetics of Diophantus			lxi
N. Progress and Proficiency of the Arabians in Algebra			lxiv
O. Communication of the Hindus with Western Nations on Astrology and Astronomy			lxxviii
BHÁSCARA.
ARITHMETIC (Lílávatí.)
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
Chapter III.	Miscellaneous.
	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
Chapter 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
Chapter V.	Progression.
	Sect. I.	Arithmetical	51
	Sect. II.	Geometrical	55
Chapter VI.	Plane Figure		58
Chapter VII.	Excavations and Content of Solids		97
Chapter VIII.	Stacks		100
Chapter IX.	Saw		101
Chapter X.	Mound of Grain		103
Chapter XI.	Shadow of a Gnomon		106
Chapter XII.	Pulverizer (Cut́t́aca)		112
Chapter XIII.	Combination		123
ALGEBRA (Víja-gańita.)
Chapter I.	Sect. I.	Invocation, &c.	129
	Sect. II.	Algorithm of Negative and Affirmative Quantities	133
	Sect. III.	Algorithm— — — of Cipher	136
	Sect. IV.	Algorithm— — — of Unknown Quantity	139
	Sect. V.	Algorithm— — — of Surds	145
Chapter II.	Pulverizer		156
Chapter III.	Affected Square. Sect. I.		170
	Sect. II.	Cyclic Method	175
	Sect. III.	Miscellaneous	179
Chapter IV.	Simple Equation		185
Chapter V.	Quadratic, &c. Equations		207
Chapter VI.	Multiliteral Equations		227
Chapter VII	Varieties of Quadratics		24.5
Chapter VIII.	Equation involving a Factum of Unknown Quantities		268
Chapter IX.	Conclusion		275
BRAHMEGUPTA.
CHAPTER XII. ARITHMETIC (Gańita.)
	Sect. I.	Algorithm	277
	Sect. II.	Mixture	287
	Sect. III.	Progression	296
	Sect. IV.	Plane Figure	305
	Sect. V.	Excavations	312
	Sect. VI.	Stacks	314
	Sect. VII.	Saw	315
	Sect. VIII.	Mounds of Grain	316
	Sect. IX.	Measure by Shadow	317
	Sect. X.	Supplement	319
CHAPTER XVIII. ALGEBRA (Cut́t́aca.)
	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