**Narayana** was the son of Nrsimha (sometimes written Narasimha). We know that he wrote his most famous work *Ganita Kaumudi* on arithmetic in 1356 but little else is known of him. His mathematical writings show that he was strongly influenced by Bhaskara II and he wrote a commentary on the *Lilavati* of Bhaskara II called *Karmapradipika.* Some historians dispute that Narayana is the author of this commentary which they attribute to Madhava.

In the *Ganita Kaumudi* Narayana considers the mathematical operation on numbers. Like many other Indian writers of arithmetics before him he considered an algorithm for multiplying numbers and he then looked at the special case of squaring numbers. One of the unusual features of Narayana's work *Karmapradipika* is that he gave seven methods of squaring numbers which are not found in the work of other Indian mathematicians.

He discussed another standard topic for Indian mathematicians namely that of finding triangles whose sides had integral values. In particular he gave a rule of finding integral triangles whose sides differ by one unit of length and which contain a pair of right-angled triangles having integral sides with a common integral height. In terms of geometry Narayana gave a rule for a segment of a circle. Narayana [4]:-

... derived his rule for a segment of a circle from Mahavira's rule for an 'elongated circle' or an ellipse-like figure.

Narayana also gave a rule to calculate approximate values of a square root. He did this by using an indeterminate equation of the second order, *Nx*^{2} + 1 = *y*^{2}, where *N* is the number whose square root is to be calculated. If *x* and *y* are a pair of roots of this equation with *x* < *y* then √*N* is approximately equal to *y*/*x*. To illustrate this method Narayana takes *N* = 10. He then finds the solutions *x* = 6, *y* = 19 which give the approximation 19/6 = 3.1666666666666666667, which is correct to 2 decimal places. Narayana then gives the solutions *x* = 228, *y* = 721 which give the approximation 721/228 = 3.1622807017543859649, correct to four places. Finally Narayana gives the pair of solutions *x* = 8658, *y* = 227379 which give the approximation 227379/8658 = 3.1622776622776622777, correct to eight decimal places. Note for comparison that √10 is, correct to 20 places, 3.1622776601683793320. See [3] for more information.

The thirteenth chapter of *Ganita Kaumudi* was called *Net of Numbers* and was devoted to number sequences. For example, he discussed some problems concerning arithmetic progressions.

The fourteenth chapter (which is the last one) of Naryana's *Ganita Kaumudi* contains a detailed discussion of magic squares and similar figures. Narayana gave the rules for the formation of doubly even, even and odd perfect magic squares along with magic triangles, rectangles and circles. He used formulae and rules for the relations between magic squares and arithmetic series. He gave methods for finding "the horizontal difference" and the first term of a magic square whose square's constant and the number of terms are given and he also gave rules for finding "the vertical difference" in the case where this information is given.

**Article by:** *J J O'Connor* and *E F Robertson*