Of intercalary months, Adhi Masas

The solar month is 30.438030202068 days

A lunar month = 29.5305881 days

It need not be added that a lunation or synodic month means the interval between two consecutive full moons or new moons. Conjunction ( New Moon ) is 0 degrees and Opposition ( Full Moon ) is 180 degrees

Hence a solar year does not have a whole number of lunar months ( about 12.37 lunations ) So a thirteenth embolismic or intercalary month is inserted.

It was observed that 19 solar years or 19*12 = 228 solar months = 235 lunations and hence 7 Adhi Masas were found in every 19 years. An intercalary or 13th month had to be inserted in a 19 year cycle and 19/7 was the ratio. .

They are called Adhi Masas in Indian Astronomy and they were computed using the Theory of continued fractions. The Theory of contiued Fractions is attributed to Euler. This 19 year old cycle is called the Metonic Cycle, named after the Greek astronomer, Meton.

But then the Indian mathematicians correctly computed the Adhi Masas, centuries before Meton or Euler ! The Indian National Calender is lunisolar, whose dates both indicate the solar year and the moon phases and the next date when the New Moon or Full Moon will occur. The length of the synodic month is given as 29.5305879 days in the Surya Siddhanta, which is correct to six decimals. Surya Siddhanta stated that there are 15933396 Adhi Masas in 51840000 solar months !

Of Intercalary months, Adhi Masas

The solar month is 30.438030202068 days

A lunar month = 29.5305881 days

It need not be added that a lunation or synodic month means the interval between two consecutive full moons or new moons. Conjunction ( New Moon ) is 0 degrees and Opposition ( Full Moon ) is 180 degrees

Hence a solar year does not have a whole number of lunar months ( about 12.37 lunations ) So a thirteenth embolismic or intercalary month is inserted.

It was observed that 19 solar years or 19*12 = 228 solar months = 235 lunations and hence 7 Adhi Masas were found in every 19 years. An intercalary or 13th month had to be inserted in a 19 year cycle and 19/7 was the ratio. .

They are called Adhi Masas in Indian Astronomy and they were computed using the Theory of continued fractions. The Theory of contiued Fractions is attributed to Euler. This 19 year old cycle is called the Metonic Cycle, named after the Greek astronomer, Meton.

But then the Indian mathematicians correctly computed the Adhi Masas, centuries before Meton or Euler ! The Indian National Calender is lunisolar, whose dates both indicate the solar year and the moon phases and the next date when the New Moon or Full Moon will occur. The length of the synodic month is given as 29.5305879 days in the Surya Siddhanta, which is correct to six decimals. Surya Siddhanta stated that there are 15933396 Adhi Masas in 51840000 solar months !

Of Vedic Maths

Consisting of 16 basic aphorisms or Sutras, Vedic Mathematics is a system of Maths which prevailed in ancient India. Composed by Bharati Krishna Thirtha, these 16 sutras help one to do faster maths.

The first aphorism is this

"Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)"

When computing the square of 9, as the nearest power of 10 is 9, let us take 10 as our base. As 9 is 1 less than 10, we can decrease it by the deficiency = 9-1 =8. This is the leftmost digit
On the right hand put deficiency^2, which is 1^2.

Hence the square of nine is 81.

For numbers above 10, instead of looking at the deficit we look at the surplus.



For example:


11^2 = (11+1)*10+1^2 = 121

12^2 = (12+2)*10+2^2 = 144

14^2 = ( 14+4)*10+4^2 = 196

25^2 = ((25+5)*2)*10+5^2 = 625

35^2= ((35+5)*3)*10+5^2 = 1225

Of Vedic Maths

Consisting of 16 basic aphorisms or Sutras, Vedic Mathematics is a system of Maths which prevailed in ancient India. Composed by Bharati Krishna Thirtha, these 16 sutras help one to do faster maths.

The first aphorism is this

"Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)"

When computing the square of 9, as the nearest power of 10 is 9, let us take 10 as our base. As 9 is 1 less than 10, we can decrease it by the deficiency = 9-1 =8. This is the leftmost digit
On the right hand put deficiency^2, which is 1^2.

Hence the square of nine is 81.

For numbers above 10, instead of looking at the deficit we look at the surplus.



For example:


11^2 = (11+1)*10+1^2 = 121

12^2 = (12+2)*10+2^2 = 144

14^2 = ( 14+4)*10+4^2 = 196

25^2 = ((25+5)*2)*10+5^2 = 625

35^2= ((35+5)*3)*10+5^2 = 1225

Mathematics and Philosophy

In India, mathematics is related to Philosophy. We can find mathematical
concepts like Zero ( Shoonyavada ), One ( Advaitavada ) and Infinity
(Poornavada ) in Philosophia Indica.

The Sine Tables of Aryabhata and Madhava, which gives correct sine values or values of
24 R Sines, at intervals of 3 degrees 45 minutes and the trignometric tables of
Brahmagupta, which gives correct sine and tan values for every 5 degrees influenced
Christopher Clavius, who headed the Gregorian Calender Reforms of 1582. These
correct trignometric tables solved the problem of the three Ls, ( Longitude, Latitude and
Loxodromes ) for the Europeans, who were looking for solutions to their navigational
problem ! It is said that Matteo Ricci was sent to India for this purpose and the
Europeans triumphed with Indian knowledge !

The Western mathematicians have indeed lauded Indian Maths & Astronomy. Here are
some quotations from maths geniuses about the long forgotten Indian Maths !

In his famous dissertation titled "Remarks on the astronomy of Indians" in 1790,
the famous Scottish mathematician, John Playfair said

"The Constructions and these tables imply a great knowledge of
geometry,arithmetic and even of the theoretical part of astronomy.But what,
without doubt is to be accounted,the greatest refinement in this system, is
the hypothesis employed in calculating the equation of the centre for the
Sun,Moon and the planets that of a circular orbit having a double
eccentricity or having its centre in the middle between the earth and the
point about which the angular motion is uniform.If to this we add the great
extent of the geometrical knowledge required to combine this and the other
principles of their astronomy together and to deduce from them the just
conclusion;the possession of a calculus equivalent to trigonometry and
lastly their approximation to the quadrature of the circle, we shall be
astonished at the magnitude of that body of science which must have
enlightened the inhabitants of India in some remote age and which whatever
it may have communicated to the Western nations appears to have received
another from them...."

Albert Einstein commented "We owe a lot to the Indians, who taught us how to count,
without which no worthwhile scientific discovery could have been made."

The great Laplace, who wrote the glorious Mechanique Celeste, remarked

"The ingenious method of expressing every possible number
using a set of ten symbols (each symbol having a place value and an absolute
value) emerged in India. The idea seems so simple nowadays that its
significance and profound importance is no longer appreciated. Its
simplicity lies in the way it facilitated calculation and placed arithmetic
foremost amongst useful inventions. The importance of this invention is more
readily appreciated when one considers that it was beyond the two greatest
men of antiquity, Archimedes and Apollonius."

Of Indian Maths

In India, mathematics is related to Philosophy. We can find mathematical
concepts like Zero ( Shoonyavada ), One ( Advaitavada ) and Infinity
(Poornavada ) in Philosophia Indica.

The Sine Tables of Aryabhata and Madhava, which gives correct sine values or values of
24 R Sines, at intervals of 3 degrees 45 minutes and the trignometric tables of
Brahmagupta, which gives correct sine and tan values for every 5 degrees influenced
Christopher Clavius, who headed the Gregorian Calender Reforms of 1582. These
correct trignometric tables solved the problem of the three Ls, ( Longitude, Latitude and
Loxodromes ) for the Europeans, who were looking for solutions to their navigational
problem ! It is said that Matteo Ricci was sent to India for this purpose and the
Europeans triumphed with Indian knowledge !

The Western mathematicians have indeed lauded Indian Maths & Astronomy. Here are
some quotations from maths geniuses about the long forgotten Indian Maths !

In his famous dissertation titled "Remarks on the astronomy of Indians" in 1790,
the famous Scottish mathematician, John Playfair said

"The Constructions and these tables imply a great knowledge of
geometry,arithmetic and even of the theoretical part of astronomy.But what,
without doubt is to be accounted,the greatest refinement in this system, is
the hypothesis employed in calculating the equation of the centre for the
Sun,Moon and the planets that of a circular orbit having a double
eccentricity or having its centre in the middle between the earth and the
point about which the angular motion is uniform.If to this we add the great
extent of the geometrical knowledge required to combine this and the other
principles of their astronomy together and to deduce from them the just
conclusion;the possession of a calculus equivalent to trigonometry and
lastly their approximation to the quadrature of the circle, we shall be
astonished at the magnitude of that body of science which must have
enlightened the inhabitants of India in some remote age and which whatever
it may have communicated to the Western nations appears to have received
another from them...."

Albert Einstein commented "We owe a lot to the Indians, who taught us how to count,
without which no worthwhile scientific discovery could have been made."

The great Laplace, who wrote the glorious Mechanique Celeste, remarked

"The ingenious method of expressing every possible number
using a set of ten symbols (each symbol having a place value and an absolute
value) emerged in India. The idea seems so simple nowadays that its
significance and profound importance is no longer appreciated. Its
simplicity lies in the way it facilitated calculation and placed arithmetic
foremost amongst useful inventions. The importance of this invention is more
readily appreciated when one considers that it was beyond the two greatest
men of antiquity, Archimedes and Apollonius."

The Infinite Pi series of Madhava

By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. ( Yukti deepika commentary )


This quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows.


c = SQRT(12 d^2 - SQRT(12 d^2/3.3 + sqrt(12 d^2)/3^2.5 - sqrt(12d^2)/3^3.7 +.......


As c = Pi d , this equation can be rewritten as


Pi = Sqrt(12( 1 - 1/3.3 + 1/3^2.5 -1/3^3.7 +......


This is obtained by substituting z = Pi/ 6 in the power series expansion for arctan (z).


Pi/4 = 1 - 1/3 +1/5 -1/7+.....


This is Madhava's formula for Pi, and this was discovered in the West by Gregory and Liebniz.