Ganitapada use area of Triangle by Aryabhata (India)

Introduce Trigonometry by Aryabhata (Ind.)

In Ganitapada 6, Aryabhata (Ind) gives the area of a triangle as:

Tribhujasya phalashriram samadalakoti bhujardhasamvargah

that translates to: “for a triangle, the result of a perpendicular with the half-side is the area.

Aryabhata (Ind) discussed the concept of sine  in his work by the name of ‘ardha-jay’ , which literally means “half-chord”. For simplicity, people started calling it jya, When Arabic writers translated his works from Sanskrit  into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning “pocket” or “fold (in a garment)”. (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona  translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means “cove” or “bay”; thence comes the English sine. Alphabetic code has been used by him to define a set of increments. If we use Aryabhata’s table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.

Indeterminate equations

A problem of great interest to Indian Mathematician group since ancient timAryabhattes has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as Diophantine equation. This is an example from mathematician Bhaskara (Ind) commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text ‘sulabha sutras’ Sulba , whose more ancient parts might date to 800 BCE. Aryabhata’s method of solving such problems is called the kuṭṭaka method. Kuttaka means “pulverizing” or “breaking into small pieces”, and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithms The diophantine equations are of interest in cryptology

To be continue ………3

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