Pramāṇa · Search · About · API

Indian Mathematics c12.s1: Appendix I. Extracts from Texts

Indian Mathematics c12.s1

Language
English

Contents

Appendix I. Extracts from Texts

The Śulvasūtras

c12.s1.t1.1

\*1.∗ In the following we shall treat of the different manners of building the *agni*. 2. We shall explain how to measure out the circuit of the area required for them.

c12.s1.t1.2

∗ \* These numbers refer to Baudhāyana's edition as translated by Dr. Thibaut.
* *L* = the Līlāvatī, *V* = Vīja Gaṇita, both by Bhāskara, *M* = Mahāvīra, *S* = Srīdhara, *C* = Chaturveda. \* *L*=the Līlāvatī, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra, *S*=Śrīdhara, *C*=Chaturveda.

c12.s1.t1.3

45. The cord stretched across a square produces an area of twice the size.

c12.s1.t1.4

46. Take the measure for the breadth, the diagonal of its square for the length: the diagonal of that oblong is the side of a square the area of which is three times the area of the square.

c12.s1.t1.5

48. The diagonal of an oblong produces by itself both the areas which the two sides of the oblong produce separately.

c12.s1.t1.6

49. This is seen in those oblongs whose sides are three and four, twelve and five, fifteen and eight, seven and twenty-four, twelve and thirty-five, fifteen and thirty-six.

c12.s1.t1.7

51. If you wish to deduct one square from another square cut off a piece from the larger square by making a mark on the ground with the side of the smaller square which you wish to deduct. Draw one of the sides across the oblong so that it touches the other side. Where it touches there cut off. By this line which has been cut off the small square is deducted from the large one.

c12.s1.t1.8

58. If you wish to turn a square into a circle draw half of the cord stretched in the diagonal from the centre towards the *prāchī* line. Describe the circle together with the third part of that piece of the cord which stands over.

c12.s1.t1.9

*Āryabhaṭa's gaṇita*—(Circa. A.D. 520).

c12.s1.t1.10

6. The area of a triangle is the product of the perpendicular common to the two halves and half the base. Half the product of this and the height is the solid with six edges.

c12.s1.t1.11

10. Add four to one hundred, multiply by eight and add again sixty-two thousand. The result is the approximate value of the circumference when the diameter is twenty thousand.

c12.s1.t1.12

13. The circle is produced by a rotation ; the triangle and the quadrilateral are determined by their hypotenuses ; the horizontal by water and the vertical by the plumb line.

c12.s1.t1.13

29. The sum of a certain number of terms diminished by each term in succession added to the whole and divided by the number of terms less one gives the value of the whole.

c12.s1.t1.14

*Brahmagupta*—(Born A.D. 598).

c12.s1.t1.15

1. He who distinctly knows addition and the rest of the twenty operations and the eight processes including measurement by shadows is a mathematician.

c12.s1.t1.16

14. The principal multiplied by its time and divided by the interest, and the quotient being multiplied by the factor less one is the time. The sum of principal and interest divided by unity added to the interest on unity is the principal.

c12.s1.t1.17

17. The number of terms less one multiplied by the common difference and added to the first term is the amount of the last. Half the sum of the last and first terms is the mean amount, and this multiplied by the number of terms is the sum of the whole.
* * * * *

c12.s1.t1.18

21. The product of half the sides and opposite sides is the rough area of a triangle or quadrilateral. Half the sum of the sides set down four times and each lessened by the sides being multiplied together—the square-root of the product is the exact area.
* * * * *

c12.s1.t1.19

40. The diameter and the square of the radius respectively multiplied by three are the practical circumference and area. The square-roots extracted from ten times the square of the same are the exact values.
* * * * *

c12.s1.t1.20

62. The integer multiplied by the sexagesimal parts of its fraction and divided by thirty is the square of the minutes and is to be added to the square of the whole degrees.
* * * * *

c12.s1.t1.21

101. These questions are stated merely for gratification. The proficient may devise a thousand others or may solve by the rules taught problems set by others.

c12.s1.t1.22

102. As the sun obscures the stars so does the expert eclipse the glory of other astronomers in an assembly of people by reciting algebraic problems, and still more by their solution.

c12.s1.t1.23

*Mahāvīra's Gaṇita-Sāra-Sangraha*—(Circa. A.D. 850).

c12.s1.t1.24

i. 13–14. The number, the diameter and the circumference of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the world, of the interspace, of the world of light, of the world of the gods and to the dwellers in hell, and miscellaneous measurements of all sorts—all these are made out by means of computation.
* * * * * vi. 147. Divide by their rate prices. Diminish by the least among them and then multiply by the least the mixed price of all the things and subtract from the given number of things. Now split up (this) into as many (as there are left) and then divide. These separated from the total price give the price of the dearest article of purchase. [This is a solution of example 36 below.]

c12.s1.t1.25

vi. 169. It has to be known that the products of gold as multiplied by their colours when divided by the mixed gold gives rise to the resulting colour (*varṇa*). [See examples 24 and 25 below.]

c12.s1.t1.26

vii. 2. Area has been taken to be of two kinds by Jina in accordance with the result—namely, that which is for practical purposes and that which is minutely accurate.

c12.s1.t1.27

vii. 233. Thus ends the section of devilishly difficult problems.

c12.s1.t1.28

*Śrīdhara’s Triśatikā*—(Circa A.D. 1030).

c12.s1.t1.29

1. Of a series of numbers beginning with unity and increasing by one, the sum is equal to half the product of the number of terms and the number of terms together with unity.

c12.s1.t1.30

32. In exchange of commodities the prices being transposed apply the previous rule (of three). With reference to the sale of living beings the price is inversely proportional to their age.

c12.s1.t1.31

65. If the gnomon be divided by twice the sum of the gnomon and the shadow the fraction of the day elapsed or which remains will be obtained.

c12.s1.t1.32

*Bhāskara*—(Born A.D. 1114).

c12.s1.t1.33

L. 1. I propound this easy process of calculation, delightful by its elegance, perspicuous with concise terms, soft and correct and pleasing to the learned.

c12.s1.t1.34

L. 139. A side is put. From that multiplied by twice some assumed number and divided by one less than the square of the assumed number a perpendicular is obtained. This being set aside is multiplied by the arbitrary number and the side as put is subtracted—the remainder will be the hypotenuse. Such a triangle is termed ‘ genuine.’

c12.s1.t1.35

L. 189. Thus, with the same sides, may be many diagonals in the quadrilateral. Yet, though indeterminate, diagonals have been sought as determinate by Brahmagupta and others.

c12.s1.t1.36

L. 213. The circumference less the arc being multiplied by the arc the product is termed ‘ first.’ From the quarter of the square of the circumference multiplied by five subtract that first product. By the remainder divide the first product multiplied by four times the diameter. The quotient will be the chord.

c12.s1.t1.37

V. 170. In the like suppositions, when the operation, owing to restriction, disappoints the answer must by the intelligent be discovered by the exercise of ingenuity. Accordingly it is said ‘ The conditions—a clear intellect, assumption of unknown quantities, equation, and the rule of three—are means of operation in analysis.’

c12.s1.t1.38

V. 224. The rule of three terms is arithmetic ; spotless understanding is algebra. What is there unknown to the intelligent ? Therefore for the dull alone it is set forth.

c12.s1.t1.39

V. 225. To augment wisdom and strengthen confidence, read, read, mathematician, this abridgement elegant in style, easily understood by youth, comprising the whole essence of calculation and containing the demonstration of its principles—full of excellence and free from defect.

Linked texts (6)