Contribution Of Mahaviracharya In The

This document summarizes the contributions of the ancient Indian mathematician Mahaviracharya (circa 850 A.D.) to the theory of series, as presented in his work Ganita Sarasangraha (GSS). The paper highlights Mahaviracharya as a significant figure, elaborating on concepts of series with lucid methods and engaging language, building upon the work of his predecessors like Aryabhata I and Brahmagupta.

The summary covers the following key areas:

• Arithmetic Progressions (AP):

  • Mahaviracharya provided detailed formulas for calculating the sum of an AP when the first term, common difference, and number of terms are known.
  • He also presented methods to find the number of terms when the first term, common difference, and sum are given.
  • The text details rules for finding the common difference and first term when other parameters are known.
  • Specific rules are outlined for "splitting up" the sum of an AP into its component elements, such as when parts of the sum are combined with the first term, common difference, or the number of terms.
  • A complex rule is presented for relating two APs, allowing for the calculation of first terms and common differences based on their sums and numbers of terms, with the possibility of sums being equal or in a specific ratio.

• Partial Sums:

  • The concept of a partial sum (the sum of a portion of a series) is explained, with a rule for calculating it.

• AP with Positive or Negative Common Difference:

  • A formula is provided for finding the sum of an AP where the common difference can be either positive or negative.

• Meeting Times of Persons with Varying Velocities:

  • Rules are given for determining when two individuals, one moving with a steady velocity and the other with velocities in an AP, will meet at a common terminus.
  • Another rule addresses the time and distance of meeting when both individuals travel with velocities in APs.

• Structures Made of Layers of Bricks:

  • A method is described for calculating the total number of bricks in layered structures, based on the number of layers and the number of bricks in the topmost layer.

• Geometric Progressions (GP):

  • Mahaviracharya provided rules for finding the "gunadhana" (a term related to the final term or a cumulative product) and the sum of a GP when the first term, common ratio, and number of terms are known.
  • An alternative, more complex rule for calculating the sum of a GP is also presented, involving a specific method of processing the number of terms.
  • The text mentions that rules for finding the last term and sum of a GP, as well as for finding the first term, common ratio, and number of terms in a GP, are also present in the GSS.
  • Rules are detailed for calculating the sum of a GP where terms are increased or decreased by a specified quantity.

• Miscellaneous Series:

  • Mahaviracharya's contribution to "miscellaneous series" is highlighted as particularly voluminous, with no other Hindu mathematician contributing as much in this area.
  • A rule is given for finding the sum of the squares of natural numbers.
  • A general rule is provided for finding the sum of the squares of numbers in an AP.
  • A rule is presented for finding the sum of the cubes of the first n natural numbers, noting it's equal to the square of the sum of the first n natural numbers.
  • A general formula is given for finding the sum of the cubes of terms in an AP.
  • A rule is outlined for calculating the sum of a series in the form of cumulative sums of natural numbers (e.g., 1 + (1+2) + (1+2+3) + ...).
  • Finally, a single, overarching formula is presented for the sum of the four types of series mentioned above (natural numbers, squares of natural numbers, cubes of natural numbers, and the cumulative sum series).

In essence, the paper underscores Mahaviracharya's profound and systematic treatment of various series, establishing him as a pioneer in elaborating on these mathematical concepts with clarity and depth.