and normalize the result:

[ sum cdots sumsum r = frac{ncdot(n+1)cdot(n+2)cdot dots cdot (n+k)}{1cdot 2 cdot 3 dots cdot (k+1)} ]

Chapter 4 is the largest chapter of the book; It contains 149 rules and 94 examples of geometric problems, including several approximation formulas for circular figures. Notable is a formula developed by Narayan to determine the area of ​​a stellar quadrilateral with the help of a so-called third diagonal.

The chordal quadrilateral ABCD resulting from interchange of sides is NS And C Tendon Squares ABED Along Diagonals F And Yes. The areas of the two quadrilaterals agree, because, according to Brahmagupta’s formula, the area depends only on the lengths of the four sides of the chord: (A = sqrt {(sa) cdot (sb) cdot (sc) ) ) With cdot(sd)})(s = frac{1}{2}cdot(a+b+c+d)) . The area of ​​the square ABED can be calculated as the sum of the areas of the sides of the triangle ABE a, c, g and triangle AED with sides b, d, g:

[A_{ABED} = A_{ABE} + A_{AED} = frac{acdot c cdot g}{4R} + frac{g cdot b cdot d}{4R} = frac{g}{4R} (ac + bd)]

On the other hand, according to Ptolemy’s theorem, the product of the lengths of the diagonals of a chordal quadrilateral is equal to the sum of the products of the opposite sides of the quadrilateral, i.e. ac + bd = him. Therefore it is as follows:

[A_{ABCD}= A_{ABED}= frac{g}{4R} (ac+bd) = frac{efg}{4R}]

One of the functions deals with special triangles whose side lengths are natural numbers and which differ by only one unit; The length of the side height of the base must also be a natural number. Narayan assumes that the left side of the base is 0.5 in lengthx – should be 2, accordingly the correct section 0.5x + 2, because the Pythagorean theorem applies to the following two partial triangles:

[(x-1)^2 – (frac{1}{2}x-2)^2 = y^2 = (x+1)^2 – (frac{1}{2}x+2)^2]

This means: (y ^ 2 = frac {3} {4} x ^ 2 -3. ) This equation has infinite solutions: (4; 3), (14; 12), (52; 45) ), (194; 168), (724; 627), …