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Mathematical Astrology III
Astronomy & Maths are inseparable from Astrology. The father of Indian Astrology, Varaha Mihira ( 500 A D ), was not only an astrologer, he was an astronomer & mathematician par excellence. The trignometric equation Sin^2 ( x ) + Cos^2 ( x ) = 1  is attributed to this Master. (The word Trignometry derives from the Sanskrit Thrikonamithi & Geometry from Jyamithi).  He wrote masterpieces in Astronomy ( Pancha Siddhantika ) & Astrology ( Horasastra ). In Astronomy and Maths, he had risen to the level of  Kepler & Laplace, only to be matched by geniuses like Aryabhata, Brahmaguptha, Bhaskara, Sir Simon Newcomb, Brown & Leverier !

He expounded the principles of Celestial Mechanics long before the great Laplace propounded them in his Mechanique Celeste.  He postulated that planetary longitudes can be found by the Threefold Trignometric Method of Manda Kriya ( Reduction to True Anomaly ), Parinathi Kriya ( Reduction to Ecliptic ) & Sheeghra Kriya ( Reduction to Perihelion) !

He was the most brilliant amongst the Nine Gems ( Nine Intellectuals), who adorned the court of King Vikramaditya. He had acquired immense spiritual powers due to Transcendental Meditation. He postulated the Helio-centric Theory of Gravitation, that all planets revolve around the Sun due to Celestial Gravity. ( Soorya Jagathah Chakshu - Guruthwakarshan ).

He postulated that the planets traverse in elliptical orbits around the Sun, that planets move faster at Sheeghrocha ( Perihelion - the point in orbit nearest to the Sun ) & slower at Mandhocha - ( Aphelion - the point in orbit farthest from the Sun), that in order to get the true longitudes of planets, one has to give Jya Samskaras ( trignometric corrections  to reduce them from Circular to Elliptical,  from Mean to True  ), long before the great Johannes Kepler discovered them in the Western World !

The Ninefold Temporal Strength  (  Kala Bala )
Kala means Time and Kala Bala means Temporal Strength or strength of time. This strength is calculated by considering the year, month, weekday, time etc of birth. The various strengths of planetary vibrations due to seasonal influences are taken into consideration and it consists of Nine Types of Strengths

1) Diurnal & Nocturnal Strength  ( Divaratri Bala )
2) Fortnight Strength  ( Paksha Bala )
3) 60 degrees Strength ( Thribhaga Bala )
4) Year Strength  (  Abda Bala )
5) Month Strength ( Masa Bala )
6) Day Strength ( Vara Bala )
7) Hour Strength ( Hora Bala )
8) Solstice Strength ( Ayana Bala )
9) Planetary War Strength  ( Yuddha Bala )
Diurnal & Nocturnal Strength  ( Divaratri Bala )

This is the strengths of planets due to diurnal or nocturnal influences. Known as Divaratri Bala, it consist of Diurnal Strength ( Diva Bala ) & Nocturnal Strength ( Ratri Bala ).

The nocturnal planets - Moon, Saturn & Mars are powerful during midnight and powerless during midday. ( Nishi Sasi Kuja Saura Sarva nJne ) and 60 Shastiamsas are assigned to them at midnight as their Divaratri strength . On the contrary, the diurnal planets - Sun, Jupiter & Venus - are powerful during midday & powerless at night and 60 Shashtiamsas are assigned to them at midday as their D R strength.  Powerful always is Mercury, be  it day or night & he  always gets 60 Shastiamsas.
Midday & Midnight

While Midday is defined as the LMN ( Local Mean Noon ), the Seers have recommended that if the birth time is marked in LMT ( Local Mean Time ), it must be converted into LAT ( Local Apparent Time) by applying the Equation of Time. The Midnight is when the Sun is in the Lower Meridian of the place & this is reckoned as 12 O Clock Midnight.

Equation for  Divaratri Bala

A day is 360 degrees and the Interval between Midday & Midnight & vice versa is 180 degrees. LMT should be converted to LAT ( Local Apparent Time ) by applying the Equation of Time. Then convert Birthtime reckoned from Midnight into degrees at 15 degrees per hour ( Ardha Rasi Bhaveth Hora ) and apply the following rule ( If birthtime in degrees exceeds 180, subtract it from 360 )

Diva Bala for Sun, Jupiter and Venus = Birth Time in Degrees / 3
Ratri  Bala of Saturn, Moon & Mars = 180 - Birth Time in Degrees / 3

in Shastiamsas

Fortnight Strength ( Paksha Bala )

Fifteen days make a Fortnight ( Paksha ) and a Paksha = 15 Lunar Days. When the Moon waxes it is Shukla Paksha ( bright half of the lunar month ) and he wanes, it is Krishna Paksha ( dark half of the lunar month ). During the Dark Half, the malefics are powerful and during the Bright Half, the benefics !

The Equation of Time

The Equation of Time = Apparent Time - Mean Time.

Two major factors, the Obliquity of the Ecliptic ( 23.45 degrees ) and the Earth's Orbital Eccentricity ( .0167 ) are responsible for the Equation of Time, which is best illustrated by an Analemma or the Figure 8 in the heavens ! The Earth moves fastest at Perihelion ( Jan 3 ) & slowest at Aphelion ( Jul 3 ).

The amplitude of the Equation of Time is 9.87 minutes.
The formula for computing the Equation of Time is
E = 9.87 Sin ( 2 B ) - 7.53 Cos (B ) - 1.5 Sin ( B)  ( E is in minutes )
where B = 360 degrees ( N - 81 )/ 364
Natural Malefics ( Papas ) & Natural Benefics ( Shubhas )
Papas are natural malefics & they are Sun, Mars, Saturn & afflicted Mercury. Shubhas are natural benefics & they are Jupiter, Venus and well associated Mercury. The waxing Moon is a benefic & the waning Moon is a malefic, viz, he is a benefic from the 8th day of the bright half of the lunar month to the 8th day of the dark half of the lunar month & a malefic in the rest of the days. The malefics ( Papas ) get more of Paksha Bala during the dark  half of the month.
Dark & Bright Fortnights ( Krishna & Shukla Pakshas )

To find out whether birth occurred during the Dark Half or the Bright Half, subtract LongSun frm LongMoon. If the difference is  < 180,  it is  the the Bright Half ( Shukla Paksha ) and if > 180 degrees, it is the Dark Half ( Krishna Paksha ).

Equations for Paksha Bala

Paksha Bala of Benefics =  LongMoon - LongSun /3
Paksha Bala of Malefics  = 60 - Paksha Bala of Benefics  (  in Shastiamsas )
The Paksha Bala of the Moon is always to be doubled.

Equations for  60 Degrees Strength ( Thribhaga Bala )

Divide Day ( Ahas ) & Ratri by 3 ( 180/3 = 60 degrees ). 60 degrees is a Thribhaga.

Mercury, Sun and Saturn rule the Thribhagas of the Day

Moon, Venus & Mars rule the Thribhagas of the Night

Find out the Thribhaga when birth has occurred. Assign 60 Shastiamsas to the ruling planet of that Thribhaga. Assign 60 Shastiamsas to Jupiter, who always gets 60 Shastiamsas

So only 2 planets gets Thribhaga Bala of 60 Shastiamsas !

So you will find that Mathematical Astrology is not easy, nor is it difficult !