Pramāṇa · Search · About · API

The Science of the Shulba: A Study in Early Hindu Geometry c19: Chapter XIV. Indeterminate Problems

The Science of the Śulba: A Study in Early Hindu Geometry c19

Language
English

Contents

Chapter XIV. Indeterminate Problems

c19.2

In the *Śulba* we find the solution of certain very interesting indeterminate problems. Some of them follow directly from the geometrical constructions made therein; others have been tacitly assumed for that purpose. We shall here notice the more important ones amongst them.

Rational Right-angled Triangles

c19.s1.1

The formula of Kātyāyana to find the sum of n equal squares of sides a each amounts to $a² (√ n)² + a² ( (n - 1)/2 )² = a² ( (n + 1)/2 )².$

c19.s1.2

Putting m² for n, in order to make the sides of the right-angled triangle free from the radical, and dividing out by a² we get $m² + ( (m² - 1)/2 )² = ( (m² + 1)/2 )² I$ as the solution of the indeterminate equation of the second degree $x² + y² = z². A$

c19.s1.3

If the sides of the right-angled triangle are to be integral as well as rational, m must be odd. According to Proclus (c. 450 A.D.), this solution was known to Pythagoras (*c.* 540 B.C.).

c19.s1.4

A still more general solution of (A) which includes the solution (I) as a particular case, is furnished by the method of transformation of a rectangle into a square. It is this: $(√mn)² + ((m-n)/2)² = ((m+n)/2)²,$ where m, n are any two arbitrary numbers. Substituting p², q² for m, n respectively, in order to eliminate the irrational quantities, we get $p² q² + ((p²-q²)/2)² = ((p²+q²)/2)². (II)$

c19.s1.5

A further generalisation has been tacitly assumed in some methods employed by Āpastamba for the construction of the *Mahāvedi*. If the sides of a rational right triangle are known, then other rational right triangles can be obtained by multiplying them by any rational integer, or how he puts it, by increasing them by any rational multiples of them. Thus if α, β, γ be a rational solution of x² + y² = z², then other rational solutions of it will be given by la, lβ, lγ where l is any rational number. This is clearly in evidence in the formula of Kātyāyana. It is now known that all rational solutions of (A) can be obtained without duplication in this way.

c19.s1.6

Karavindasvāmī gives the solution,¹ $x, ((m²+2m)/(2m+2))x, ((m²+2m+2)/(2m+2))x.$ It can of course be derived from the solution (I) by multiplying by x and dividing by m² every element of it and then putting m+1 for m.

c19.s1.7

¹ Vide his commentary on *ĀpŚl*, i. 4.

c19.s1.8

*Rational Right Triangles having a Given Leg.*

c19.s1.9

There seems to have been an attempt to find the rational right triangles having a given leg. We find in the *Śulba* at least two such right triangles having a common leg a, *viz.*, $(a, 3a/4, 5a/4) and (a, 5a/12, 13a/12).$

c19.s1.10

The principle underlying these solutions will be easily detected to be that of the reduction of the sides of any rational right triangle in the ratio of the given leg to the corresponding side of it. Thus the sides of a rational right triangle having a given leg a will be (a, aβ/α, aγ/α), where α, β, γ are the sides of any rational right triangle. So that starting with the solution (II), we shall find that all rational right triangles having the leg a will be given by $a, ((p²-q²)/2pq)a, ((p²+q²)/2pq)a.$ But this general solution is not stated anywhere in the *Śulba*. The commentators of Āpastamba, however, give the solution $a, ((m²+2m)/(2m+2))a, ((m²+2m+2)/(2m+2))a.$

c19.s1.11

It should be noted that the above principle for obtaining the rational right triangles having a given leg has been followed expressly in later times in India by Mahāvīra (850) and in Europe by Leonardo Fibonacci of Pisa (1202) and Vieta (*c.* 1580).

Simultaneous Indeterminate Equations

c19.s2.1

The construction of the *vedi* gives rise to a type of problems of indeterminate character, though in a few cases the physical conditions are so prescribed as to make them determinate.¹ For instance, the breadth of the *Gārhapatya* *vedi* shall have to be, according to the tradition of the scriptures, one vyāyāma.¹ But its form should be square according to some and circular according to others.² It is to be constructed with five layers of bricks, each layer consisting of 21 bricks. Further the rifts of bricks in two consecutive layers must not coincide. The shape of the bricks employed may be square or rectangular. Now the most interesting case is that in which the *vedi* and also the bricks employed in constructing it, are square in shape. It is said that three kinds of square bricks should be made with the sixth, fourth and third part of a vyāyāma as a side. The first layer (l₁) should be constructed with 9 bricks of the first kind and 12 bricks of the second kind, and the second layer (l₂) with 5 bricks of the third kind and 16 bricks of the first kind.³ That is, if we denote the bricks of different kinds thus: b₁ = v²/36, b₂ = v²/16, b₃ = v²/9, where v denotes a vyāyāma, then $l₁ = 9b₁ + 12b₂, l₂ = 5b₃ + 16b₁.$

c19.s2.2

¹ See Bibhutibhusan Datta, "The Origin of Hindu Indeterminate Analysis," *Archeion*, XII (1931), pp. 401-407.

c19.s2.3

¹ *BŚl*, ii. 61; *ĀpŚl*, vii. 5; compare also *ĀpŚl*, vii. 10.

c19.s2.4

² *BŚl*, ii. 62-3; *ĀpŚl*, vii. 6.

c19.s2.5

³ *BŚl*, ii. 66-9.

c19.s2.6

The third and fifth layers are replica of the first and the fourth, of the second.

c19.s2.7

Now it may be asked how the ancient altar-builders determined the size of the bricks of different kinds and the number of bricks of each kind that will be required for the construction of each layer. They proceeded probably in some way like this: Since 21 is not a square number, no layer of the altar can be constructed with bricks of the same kind. So the number of kinds of bricks employed in any layer must be at least 2. Since no two successive layers should have identical cleavage, all the bricks employed in the second layer must not be the same as those of the first layer. Hence the minimum kinds of bricks must be 3. Assume then their sides to be *p*, *q* and *r*th part of a vyāyāma, where *p*, *q*, *r* are rational integers to be determined. Suppose the first layer consists of *x* bricks of the first kind and *y* bricks of the second kind. Then we must have $x/(p²) + y/(q²) = 1,$ $x + y = 21.$

c19.s2.8

Similarly if the second layer consists of *u* bricks of the third kind and *v* bricks of the first kind, $u/(r²) + v/(p²) = 1$ $u + v = 21.$

c19.s2.9

Thus we are led to the simultaneous indeterminate equations .x/(m²) + y/(n²) &= 1 / x + y &= 21.} \dots (A)

c19.s2.10

Baudhāyana's statements about the size of the different varieties of bricks and the number of them employed in the construction of a particular layer, amount to the following solutions of the equations (A) .m &= 6, x = 9, / n &= 4, y = 12;} .m &= 3, x = 5, / n &= 6, y = 16.}

c19.s2.11

These solutions were probably obtained by trial in succession, thus: Solving (A) as simultaneous linear equations in *x* and *y*, we get $x = (m²(21-n²))/(m²-n²),$ $y = (n²(m²-21))/(m²-n²).$

c19.s2.12

Now the physical circumstances of the problem are such that x and y must be positive integers. Therefore if m > n $m² > 21 > n²$ or $m > √21 > n.$ Since $5 > √21 > 4$ therefore we must have $m ≥ 5, n ≤ 4.$ If on the contrary m < n, we must have $n² > 21 > m²$ Hence $n ≥ 5, m ≤ 4.$

c19.s2.13

Then substituting in the expression for x one after another the values n = 4, 3, 2, 1, we can determine by trial the value of m which will make the value of x in each case integral. Thus we shall easily arrive at the solutions given by Baudhāyana.

c19.s2.14

A much more difficult problem of the same type arises in connexion with the construction of the falcon-shaped Fire-altar. Its total area is given to be 7½ a², where a = a puruṣa. It is laid down that the altar must be constructed in fire layers and the number of bricks employed in any layer must be 200. As before, the rifts of bricks in any two successive layers must not coincide. There is no injunction of the scriptures about the varieties of the bricks to be employed or about their relative size and shape. In one method of construction Baudhāyana employs four kinds of square bricks whose sides are respectively the fourth, fifth, sixth and tenth part of a. Let us take, in general, the areas of bricks to be a²/m, a²/n, a²/p, a²/q. If x, y, z, u be the number of bricks of each variety respectively that are employed in a layer, the problem amounts algebraically to the solution of the indeterminate equations $.x/m+y/n+z/p+u/q &= 7½, / x+y+z+u &= 200. (B)$ Baudhāyana gives two integral solutions of these equations :¹

c19.s2.15

¹ *BŚl*, iii. 24 ff.

c19.s2.16

(*i*) m=16, n=25, p=36, q=100
(*i*·1) x=24, y=120, z=36, u=20
*or*
(*i*·2) x=12, y=125, z=63, u=0.

c19.s2.17

Since in this method the values of m, n, p, q are perfect squares. the shape of all the bricks are square. Baudhāyana has described a second method of construction of the *Agni* in which he employs certain rectangular bricks too. These bricks, it may be noted, are easily divisible into square shapes. But as that will increase the number of bricks employed in the construction of the *Agni* he has refrained from doing so. All those things are, however, immaterial for us who look upon his problem from the point of view of the solution of algebraic equations. Baudhāyana's new solutions of the equations (B) are²

c19.s2.18

² *BŚl*, iii. 41 ff.

c19.s2.19

(*ii*) m=25, n=50, p=50/3, q=100
(*ii*·1) x=160, y=30, z=8, u=2
*or*
(*ii*·2) x=165, y=25, z=6, u=4

c19.s2.20

Āpastamba uses square bricks of five different varieties for the construction of the same *Agni*. So his problem will be represented algebraically by the equations $.x/m+y/n+z/p+u/q+v/r &= 7½, / x+y+z+u+v &= 200 ; (C)$ where *m*, *n*, *p*, *q*, *r* are perfect squares. Āpastamba's statement of his method of solution, partly explained in detail and partly hinted, is not free from ambiguity.¹ Consequently it has been interpreted differently by his different commentators leading consequently to several solutions. According to Karavindasvāmī, the two solutions of (C) will be

c19.s2.21

¹ *ĀpŚl*, xi. 1 ff. The text describing one solution is positively faulty as has also been noticed by all the commentators. Bürk failed to detect this. He thinks erroneously that the preliminary arrangement for the second construction is comprised of 194 bricks, whereas it actually consists of 198 bricks (*vide* *ZDMG*, LVI, p. 366).

c19.s2.22

$(i) m = 16, n = 25, p = 64, q = 100, r = 144, / x = 67, y = 58, z = 48, u = 18, v = 9;$

c19.s2.23

$(ii) m = 16, n = 25, p = 36, q = 64, r = 100, / x = 12, y = 157, z = 9, u = 0, v = 22.$

c19.s2.24

To these Kapardisvāmī adds the solution

c19.s2.25

$(iii) m = 16, n = 25, p = 36, q = 64, r = 100, / x = 10, y = 159, z = 9, u = 8, v = 14.$

c19.s2.26

Sundararāja's interpretation leads to as many as four new solutions of (C)

c19.s2.27

$(iv), (v) m = 16, n = 25, p = 36, q = 64, r = 100, / x = 70, 12; y = 45, 157; z = 9, u = 56, 0; v = 20, 22;$

c19.s2.28

$(vi), (vii) m = 16, n = 25, p = 64, q = 100, r = 144, / x = 74, 77; y = 45, 42; z = 52, 40; u = 20, 32; v = 9$

c19.s2.29

He has also added a few more solutions of his own. All these show very clearly that the Hindus fully recognised the indeterminate character of the above problem of the construction of the Fire-altar of the shape of the falcon.

c19.s2.30

Several other indeterminate problems of the above type present themselves in connexion with the construction of the Fire-altars of other shapes. We need not dilate upon them here. It should, however, be noted that the actual difficulties of construction are much more than what will appear from the mere algebraic considerations. For the bricks will have to be arranged in such a configuration as to have the prescribed shape of the altar.

Linked texts (2)