C Program to predict Poorna Chandra using Brahmagupta's Method

Here we will gain a good understanding how the algorithm works...

Ujjain

Recently I had a opportunity to visit Ujjain.I was really amazed to learn about astronomical significance of Ujjain.I decided to dive deeper into Mathematics which is a important component of astronomical sciences.

Some interesting astronomical Facts about Ujjain

1. The Tropic of Cancer crosses the spire of Mahakal Temple in Ujjain, as it does the Somnath Temple in Gujarat. This imaginary line is also said to pass through Ujjain’s temple of Mangalnath, considered in Hindu cosmogony to be the birthplace of Mangal (Mars) and the closest point from Earth to Mars.
2. As per the Surya Siddhanta, a 4th-century astronomical treatise, Ujjain is geographically situated at the precise spot where the zero meridian of longitude and the Tropic of Cancer intersect. This is why it was considered the navel of the earth, and is called the “Greenwich of India”.
3. In ancient times, astronomers like Varahamihira, Brahmagupta, and Bhaskaracharya among other luminaries, also made Ujjain their home. Additionally, it was here in Ujjain that the legendary king Vikramaditya drove away the Sakas and started a new era, the Vikram Samvat or the Ujjain calendar, around 58–56 B.C.In this edition we will be exploring one of the interesting Mathematical Method by Brahmagupta which have a application in astronomy with the help of a C program.

Brahmagupta

Ujjain was the centre of Ancient Bharatiya mathematical astronomy. Brahmagupta was the director of this centre. Brahmagupta wrote many textbooks for mathematics and astronomy while he was in Ujjain. These include ‘Durkeamynarda’ (672), ‘Khandakhadyaka’ (665), ‘Brahmasphutasiddhanta’ (628) and ‘Cadamakela’ (624). The ‘Brahmasphutasiddhanta’ meaning the ‘Corrected Treatise of Brahma’ is one of his well-known works.

Brahmasphutasiddhanta

This book on mathematical astronomy contains a substantial amount of mathematical material written in c. 628

Kuttākāra

kuttaka denotes a kitchen utensil used for breaking things into small pieces.A very interesting application of the kuttaka can be made in adjusting the calendar.The Kuttakādhyāya of Brāhmasphutasiddhānta consists of about 100 verses.

Chapters covered in Kuttukadhyaya

Let us understand the Algorithm with implementation

Before this let us understand few basic astronomical things which we will use to predict poorna chandra year using Kuttuka

The phenomenon of 19-year eclipse cycle is called the ‘Metonic cycle’.For every 19 years that go by, almost 235 lunar months come to pass as well. The keyword being ‘almost’. The 235 lunar months fall short by 72 minutes. When they add up, they push the Earth out of sync with the solstice and the cycle needs to start over.If we take a unit of time 1/235 of a solar year (about 37 hours, 18minutes), then a year is 235 units long and a lunar month is 19 units long.

So, on January 27 1994 there was a full Moon we have to find out what is the next year after 1994 in which the moon will be full on February 19?

There are 23 days between Jan 27 and 19 Feb which is almost 15 of our basic time units.

How?

1 lunar month is almost 29.5 days which is 19 units..

So,how much will 1 day will correspond to 0.644

So how many units will 23 days will correspond to approximate 15

Now using Kuttuka find we will smallest value of x and y,where x is months and y is the years..

19 x = 235y + 15

Diaphantine Equations

A linear Diophantine equation (in two variables) is an equation of the general form

ax+by=c

Such equations can be solved completely, and the first known solution was constructed by Brahmagupta. Consider the equation

19 x = 235y + 15

Place the Quotients as shown in the pattern above.

If the number of quotients is even, put rb below them instead of -rb.

The penultimate has to be multiplied with the one above and added to the ultimate.

How we will complete the Valli further....

If we subtract 235 from 1485 repeatedly till it dont becomes 0 , we will get smallest positive integer value of x i.e x=75 which occurs at k=-6

So, y will be 6..

Interesting , the solution of linear Diophantine equations is not difficult. The only disadvantage to the method used above is the tedious trial-and-error procedure of getting one solution. That is where the method called the kuttaka(pulverizer) comes in. This technique shortens the labor of finding the first solution by a considerable amount

1994+6=2000

So after 6 years from 27 Januray 1994 on February 19,2000 there will be a full moon.

Let us check..

The moon was indeed full on February 19,2000.

That's all for this month's edition , in next edition we will be discussing a c program for this with the help of some interesting visualization...