The Science of the Shulba: A Study in Early Hindu Geometry c20.s5: Approximate Formula
The Science of the Śulba: A Study in Early Hindu Geometry c20.s5
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Approximate Formula
c20.s5.1
It would be natural to ask if there is in the *Śulba* any specific rule for determining the approximate value of any surd. But the answer is not very reassuring. For we do not find an express statement of any formula for the purpose. At the same time we can unhesitatingly admit that they had the necessary equipments for the approximate evaluation of surds, at least of some. We have pointed out before that in the science of the *Śulba*, it is sometimes necessary to construct a square having a given area. And that is a geometrical method of finding the square-root of a given number. If the given area is represented by a non-square number, we get a method finding the square-root of a non-square number. One formula follows at once from the method given in the *Śulba*, for the transformation of a rectangle into a square, which has been described before. According to this method, a square (having its side equal to the breadth of the given rectangle) is first cut off from the given rectangle; the excess portion is divided into two halves which are then joined to the two sides of that square. Then by adding a small square in the corner, a large square is completed. The square equivalent to the given rectangle will be obtained, it is said, by subtracting the added small square from the completed large square. This subtraction was doubtless made by the Śulba-workers with the help of the theorem of the square of the diagonal. But it can also be made by cutting off two strips from the two sides, say, the east side and the south side of the completed large square.¹ Suppose A to be the area of the given rectangle and a be the side of the square subtracted from it. Then the side of the large completed square will be a + (A-a²)/2a. The area of the small added square will be ((A-a²)/2a)². Then from each side of the large sphere we shall have to cut off a thin strip of the same breadth. If x denote the breadth of the strip, we must have
c20.s5.2
¹ The process is in fact the reverse of that taught in the *Śulba* for the increment of a given square into another square.
c20.s5.3
$((A-a²)/2a)² = 2x(a + (A-a²)/2a) - x²$
c20.s5.4
$or x = (((A-a²)/2a)²)/(2(a + (A-a²)/2a)),$
c20.s5.5
neglecting x² as being very small. Hence we finally arrive at the formula
c20.s5.6
$√ A = a + (A-a²)/2a - (((A-a²)/2a)²)/(2(a + (A-a²)/2a))$
c20.s5.7
This formula requires a correction, it will be easily recognised, inasmuch as a portion equivalent to x² has been subtracted too much.
c20.s5.8
The rule of the Bakhshālī Manuscript for determining the approximate root of a non-square number must have been obtained exactly in this way. It says ¹
c20.s5.9
¹ Bibhutibhusan Datta, "The Bakhshālī Mathematics," *Bull. Cal. Math. Soc.*, XXI, 1929, pp. 1-60.
c20.s5.10
"In case of a non-square (number), subtract the nearest square number; divide the remainder by twice (the root of that number). Half the square of that (that is, the fraction just obtained) is divided by the sum of the root and the fraction and subtract; (this will be the approximate value of the root) less the square (of the last term)."
c20.s5.11
If A = a² + r, we write the above formula as
c20.s5.12
$√A = √(a² + r) = a + r/2a - ((r/2a)²)/(2(a + r/2a)).$
c20.s5.13
Now this formula will not be available for finding the approximate values of those surds in which r is not small compared with a². So Rodet ² holds that a different process of approximation to the value of a surd was also known to the authors of the *Sulba*. It will lead to the formula, says he,
c20.s5.14
$√(a² + r) = a + r/(2a + 1) + (r/(2a + 1)(1 - r/(2a + 1)))/(2(a + r/(2a + 1))) + ε$ where $ε = [ r - r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1))) .$ $. × 2a + r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1))) ]$ $÷ 2 a + r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1)))$
c20.s5.15
² L. Rodet, "Sur une méthode d'approximation des recines cares conne dans l'Inde antérieurment a la conquite d'Alexandre, *Bull. Soc. Math. d. France*, VII, 1879, pp. 98-102; "Sur les méthodes d'approximation chez las anciens."--*Ibid*, pp. 159-167.
c20.s5.16
A nearly equivalent formula will be obtained by proceeding in the following way: From the given rectangle A, cut off the square (S) of side a each. Let the area of the remaining portion of the rectangle be r. From this cut off two strips I and II of the same breadth r/(2a+1) and arrange them as in Fig. 78. Then the area of the strip of the rectangle still left over will be $= r - 2a × r/(2a+1) = r/(2a+1).$
c20.s5.18
Complete the larger square by adding a portion III which is deducted from r/(2a+1). Then the area of IV will be $= r/(2a+1) - (r / a+1)² = r/(2a+1)(1 - r/(2a+1)).$
c20.s5.19
Now this portion can be utilised in increasing the side of the square obtained before. If the increment of the side be y, then we must have $2y(a + r/(2a+1)) + y² = r/(2a+1)(1 - r/(2a+1)).$
c20.s5.20
Therefore, approximately $y = (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1)))$
c20.s5.21
Hence the side of the equivalent square is nearly $a + r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1))).$
c20.s5.22
This is a little too great. So decrease the square by cutting off a strip of breadth ε from either side; then $2ea + r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1))) - ε²$ $= (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1)))²$ whence, we get
c20.s5.23
\epsilon = { (r/(2a+1)(1-r/(2a+1)))/(2(a+r/(2a+1))) }²
c20.s5.24
\div 2{a+r/(2a+1)+(r/(2a+1)(1-r/(2a+1)))/(2(a+r/(2a+1)))}
c20.s5.25
Thus we have finally
c20.s5.26
\surd A = \surd a²+r = a+r/(2a+1)+(r/(2a+1)(1-r/(2a+1)))/(2(a+r/(2a+1)))-\epsilon
c20.s5.27
nearly.
Linked texts (1)
- Bakhshālī Manuscript — Sanskrit