Square root of '2' in the शुल्बसूत्रs.
Arvind_Kolhatkar
Square root of '2' in the शुल्बसूत्रs.
Among the various geometrical procedures laid down in the शुल्ब सूत्रs is the one for giving the approximate value of the square-root of the number ‘2’. This value is needed when a वेदि, square in shape, whose area is twice that of another square वेदि, has to constructed. If a square-shaped वेदि has its side equal to 1 पाद, that is 12 अङ्गुलs, the शुल्बसूत्रकारs knew that the measure of its hypotenuse will be square-root of 2. In this context बौधायन says: समचतुरस्रस्याक्ष्णयारज्जुर्द्विस्तावतीं भूमिं करोति। बौधायन १.९. The hypotenuse – अक्ष्णयारज्जु – of a square – समचतुरस्र – creates twice the area. If a square be drawn with its side as the hypotenuse of another square, the area covered by it will be equal to 2 times the area covered by the smaller square.
This is a special case of another सूत्र which says that the hypotenuse of a rectangle creates a square equal in area to the sum of the squares on the two sides of that rectangle. (दीर्घचतुरस्रस्याक्ष्णयारज्जु: पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति। तासां त्रिकचतुष्कयोर्द्वादशिकपञ्चिकयो: पञ्चदशिकाष्टिकयो: सप्तिकचतुर्विंशिकयोर्द्वादशिकपञ्चत्रिंशिकयो: पञ्चदशिकषट्त्रिंशिकयोरित्येतासूपलब्धि:। बौधायन १.१२. (The hypotenuse of a rectangle creates a square equal to that created separately by the vertical and transverse sides of the rectangle. This is seen in rectangles of 3,4; 12,5; 15,8; 7,24; 12,35; 15,36 etc.)
A little effort would show that no matter how much one tries, it is not possible to find an exact measure of the square-root of 2. No number expressible as the quotient of two integers or whole numbers can be found that exactly equals the square-root of 2. We have to remain satisfied with an approximation, sufficiently refined to meet the task in hand.
The शुल्बसूत्रकारs gave their own approximate value of square-root of 2. All शुल्ब सूत्रs, in words slightly different from each other, give the same formula. This is the formula that बौधायन gives:
प्रमाणं तृतीयेन वर्धयेत्तच्च चतुर्थेनात्मचतुस्त्रिंशोनेन। सविशेष:। बौधायन २.१२.
Increase the measure by its third, that by its fourth and of that reduce its thirty-fourth and add a small difference. (This is the side of a square whose area is twice that of a given square.)
Expressed in numbers, this says that if the side of the given square is equal to 1, the side of the desired square of double the area is 1+1/3+1/3×4 -1/3×4×34. This number, calculated in decimal fractions, is equal to 1·4142156… The same value, calculated by modern methods is 1·414213… . How did the शुल्बसूत्रकारs, or whoever found this, discovered the value of the square-root of 2 so remarkably close to the modern one, without the assistance of techniques of later ages, such as the concept of Zero, the four basic arithmetical operations and the number system that we now use?
George Thibaut has, in his book on शुल्बसूत्रs, attempted to re-create one possible way in which this could have been done. He starts with writing down groups of numbers as under:
(1,1,2,1), (2,4.8,9), (3,9.18,16), (4,16,32,36), (5,25,50,49), (6,36,72,64), (7,49,98,100), (8,64,128,121), (9,81,162,169), (10,100,200,196), (11,121,242,225), (12,144,288,279), (13,169,338,324), (14,196,392,400), (15,225,450,441), (16,256,512,529), (17,289,578,576), (18,324,648,625), (19,361,722,729), (20,400.800,784) and so on. In these groups, the second number is the square of the first, the third number is that square doubled and the last number is the nearest square number. Inspection will show that in the groups (2,4.8,9), (5,25,50,49) and (12,144,288,279) the doubled square differs from the nearest square by 1. Thus, in (2,4.8,9), we see that the hypotenuse of a square of side 2 is close to 3, in (5,25,50,49) the hypotenuse of a square of side 5 is close to 7 and in the group (12,144,288,279), the hypotenuse of a square of side 12 is close to 17. This series can be extended as far as one desires but the शुल्बसूत्रकारs have stopped with (12,144,288,279). (Here a question might be asked how, in the absence of the knowledge of arithmetical operations, could the शुल्बसूत्रकारs found these values of squares, which requires knowing how to multiply one number with another. The answer is that the square of a number can be found without the knowledge of multiplication. For this, arrange pebbles, as many as that number, in a line. Repeat this as many times as that number and then count the pebbles in the square thus formed.)
Thus it is seen that a square with side equal to 17 has an area of 289, which is 1 more than double the area of another square with side 12. If the sides of the larger square can be shortened so that the reduction in the area is equal to 1, the shortened square will have the side whose square is exactly double that of the square with side 12. To understand how this can be done, look at the diagram shown below.
In the figure, ABCD is the square with side 17 and area 289. Two equal strips EJCD and AHGD are cut out from its पार्श्वमानी and तिर्यङ्मानी sides such that the combined area of these strips will be equal to 1. Therefore the area of the shortened square HBJF should be equal to 288 and its side equal to the hypotenuse of the square with side 12. It follows that for this to happen, the width of the cut-out strips has to be 1/34, so that the areas of each strip will be 17×1/34 = 1/2, and the total of areas of these two strips will be =1. The only problem that arises here is that in this calculation, the overlapped portion of these strips, viz., EFGD has been counted twice, while in reality is can be counted only once. In other words, the area of the square HBJF is 288 less the area of EFGD and the side of the desired square is almost 17 - 1/34 and a small difference. This can be written as 12+4+1-1/34 =12(1+1/3+1/3×4 -1/3×4×34.) and a little difference or सविशेष. (This method has been explained in greater detail at http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html)
I can think of an easier way of explain this. Thibaut had to tediously work his numbers by using pencil and paper. By using Excel, I can create in a matter of seconds a worksheet in which column 1 has a number, column 2 has its square, column 3 has twice the square and column 4 has the square-root of the number in column 3. I created such a worksheet for numbers 1 to 700 and visually examined it to locate where in column 4 is the number closest to an integer (a whole number). At 408, double its square is 332,928 and its square-root is 576.9991. This differs from the nearest integer 577 by just 0.0009. No other number in the first 700 numbers gives such a close fit. It means that if the side of the original square is 408 units, the area of a square whose side is 577 is very close to 2 times the area of the first square.
How do we derive 577, starting from 408? We do so by repeated approximations. Write:
577=408+136+34-1=408(1+1/3+1/3×4 -1/3×4×34.)
The units of measurement in शुल्बसूत्रs are पाद, अङ्गुल and यव and their relationship is 1 पाद = 12 अङ्गुलs and 1 अङ्गुल = 34 यवs. We can now see why the awkward number of 34 यवs has been chosen to be equal to 1 अङ्गुल. Normally more user-friendly numbers, such as 4,5,10,25,8,16,32 are chosen in such tables. The choice of 34 is explained when we see that it occurs naturally in the above working. 1 पाद then becomes 408 यवs.
Arvind Kolhatkar, Toronto, March 18, 2014.
Arvind_Kolhatkar
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Arvind_Kolhatkar
G S S Murthy
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