Dasha Mathematics
दशा गणित — The Algebra of Planetary Time
Vedic Jyotish is not mysticism — it is a precise mathematical system for mapping cosmic time onto human life. The Vimshottari Dasha divides 120 years into nine planetary periods based on the Moon's Nakshatra at birth. Ashtakavarga scores the benefic potential of every house with an algorithm involving 7 planets and the lagna. Shodashamsha divides each degree into 16 parts — precision to 1°52'30" of arc. This is mathematics in service of understanding the self.
Vimshottari Dasha — The 120-Year Cycle
Vimshottari (Sanskrit: “120”) Dasha is the most widely used planetary period system. The 120-year cycle is divided among 9 planets (Grahas + Nodes) in a fixed sequence based on the 27 Nakshatras. Each Nakshatra group of 3 belongs to one planet. Your birth Moon Nakshatra determines where in the sequence you begin — and at what fraction through that planet's period.
The Key Formula
Dasha Balance at Birth:
B = Dlord × (1 − elapsednakshatra / spannakshatra)
Where:
Dlord = total years of Dasha lord
elapsednakshatra = degrees Moon has traversed in birth Nakshatra (0–13°20')
spannakshatra = 13°20' = 800 arcminutes
Example: Moon at 5° Ashwini (Ketu Nakshatra, 13°20' span)
B = 7 × (1 − 5/13.33) = 7 × 0.625 = 4.375 years remaining in Ketu Dasha
Antardasha (Sub-Period) Formula
AntardashaX in Y = (YearsX × YearsY × 12) / 120 months
Example: Moon Antardasha within Saturn Mahadasha
= (10 × 19 × 12) / 120 = 19 months = 1 year 7 months
The Nine Dasha Lords — Total: 120 Years
South Node — dissolution, moksha · starts at age 0
Beauty, arts, relationships · starts at age 7
Soul, authority, father · starts at age 27
Mind, mother, emotions · starts at age 33
Energy, siblings, property · starts at age 43
North Node — worldly obsession · starts at age 50
Wisdom, teachers, dharma · starts at age 68
Karma, discipline, delay · starts at age 84
Intellect, commerce, communication · starts at age 103
Saturn Mahadasha (19 years) — Complete Antardasha Breakdown
| Antardasha Lord | Duration | Formula |
|---|---|---|
| Ketu | 1y 1m 9d | (19 × 7 × 12) / 120 = 13.30 months |
| Venus | 3y 2m 0d | (19 × 20 × 12) / 120 = 38.00 months |
| Sun | 0y 11m 12d | (19 × 6 × 12) / 120 = 11.40 months |
| Moon | 1y 7m 0d | (19 × 10 × 12) / 120 = 19.00 months |
| Mars | 1y 1m 9d | (19 × 7 × 12) / 120 = 13.30 months |
| Rahu | 2y 10m 6d | (19 × 18 × 12) / 120 = 34.20 months |
| Jupiter | 2y 6m 12d | (19 × 16 × 12) / 120 = 30.40 months |
| Saturn | 3y 0m 3d | (19 × 19 × 12) / 120 = 36.10 months |
| Mercury | 2y 8m 9d | (19 × 17 × 12) / 120 = 32.30 months |
| Total | 19 years | = 228 months exactly |
Ashtakavarga — The Eight-Source Scoring System
Ashtakavarga (Sanskrit: “eight divisions”) is a system that computes the benefic strength of each of the 12 houses through contributions from 7 planets + the Ascendant (Lagna) = 8 sources. Each source contributes 0 or 1 “bindu” (point) to each house based on precise positional rules. A house with 4+ bindus is strong; 0–3 is weak. The total across all 12 houses for one planet is called Sarvashtakavarga — used to time transits and dashas.
The Algorithm
For each planet P and each source S (planet or Lagna):
Bindu(P, house H) += 1
if house H falls in a “friendly” position from S
as defined by the Ashtakavarga table
Sarvashtakavarga score for house H:
SAV(H) = Σ Bindu(P, H) for all 7 planets
Range: 0 to 56 (7 planets × max 8 bindus each)
Typical strong house: SAV ≥ 28
Transit Strength via Ashtakavarga:
Tstrength = BAV(transiting planet in house H)
4+ bindus → beneficial transit
0–3 bindus → difficult transit period
Sun gives bindus from its own position and from other planets in specific house positions. Maximum 8 bindus.
Benefic houses from Sun: 1, 2, 4, 7, 8, 9, 10, 11
Moon is weak in Ashtakavarga — only 4 possible bindus, making emotional sectors harder to fill.
Benefic houses from Moon: 3, 6, 10, 11
Mars bindus indicate competitive strength, energy deployment zones.
Benefic houses from Mars: 3, 5, 6, 10, 11
Mercury has the highest bindu potential — intellect and communication flow abundantly.
Benefic houses from Mercury: 1, 3, 5, 6, 9, 10, 11, 12
Jupiter expands the houses it aspects — 5 bindus indicate where dharma and abundance flow.
Benefic houses from Jupiter: 2, 5, 7, 9, 11
Venus has maximum 9 bindus — the most benefic planet in Ashtakavarga analysis.
Benefic houses from Venus: 1, 2, 3, 4, 5, 8, 9, 11, 12
Saturn is restricted — only 4 bindus, confirming its karmic limiting principle.
Benefic houses from Saturn: 3, 5, 6, 11
Trikona Shodhana — The Reduction Algorithm
Raw Ashtakavarga scores are further refined by Trikona Shodhana (triangular reduction) and Ekadhipatya Shodhana (single-lord reduction). The algorithm:
Step 1 — Trikona Shodhana:
Find minimum of 3 trikona houses: min(H1, H5, H9) for 1st trikona
Subtract this minimum from all three: H1 -= min; H5 -= min; H9 -= min
Repeat for H2/H6/H10 and H3/H7/H11 and H4/H8/H12
Step 2 — Ekadhipatya Shodhana:
For each lord ruling 2 houses: keep higher-scored house, zero out lower
Exception: if one house contains the planet, zero out the other regardless
Result: Prastara Ashtakavarga → final, refined strength map
16 Divisional Charts — The Varga System
Each zodiac sign spans 30°. Parashara describes 16 primary divisional charts (vargas), each dividing each sign into progressively smaller subdivisions. The mathematics: a D-n chart divides each sign into n equal parts of (30/n)° each. This allows precision to fractions of a degree.
The Divisional Chart Formula
D-n position calculation:
offset = position_within_sign (0° to 30°)
division = floor(offset × n / 30)
D-n_sign = (sign_index × n + division) mod 12
Example: Planet at 14°20' Aries for D-9 (Navamsha):
offset = 14.33°, n = 9
division = floor(14.33 × 9 / 30) = floor(4.3) = 4
D-9 sign = (0 × 9 + 4) mod 12 = 4 = Leo
| Chart | Division | Arc Size | Signification |
|---|---|---|---|
| Rashi (D-1) | D-1 | 30° | Lagna chart — the whole life. The foundation of all analysis. |
| Hora (D-2) | D-2 | 15° | Wealth, financial potential. Sun/Moon hora for day/night planets. |
| Drekkana (D-3) | D-3 | 10° | Siblings, courage, short journeys, valour in action. |
| Chaturthamsha (D-4) | D-4 | 7°30' | Property, home, vehicles, fixed assets. |
| Saptamamsha (D-7) | D-7 | 4°17' | Children, progeny, creative self-expression. |
| Navamsha (D-9) | D-9 | 3°20' | Dharma, marriage, spiritual merit. The most important divisional chart after D-1. |
| Dashamsha (D-10) | D-10 | 3° | Career, profession, social contribution, public reputation. |
| Dvadashamsha (D-12) | D-12 | 2°30' | Parents, ancestry, karma from previous generation. |
| Shodashamsha (D-16) | D-16 | 1°52'30'' | Vehicles, pleasures, mobile assets, travel comfort. |
| Vimshamsha (D-20) | D-20 | 1°30' | Spiritual practice, upasana, religious rituals. |
| Chaturvimshamsha (D-24) | D-24 | 1°15' | Education, learning, vidya, academic achievement. |
| Saptavimshamsha (D-27) | D-27 | 1°6'40'' | Strength, vitality, physical endurance (bala). |
| Trimshamsha (D-30) | D-30 | 1° | Evil, misfortune, health crises, character flaws. |
| Khavedamsha (D-40) | D-40 | 0°45' | Auspicious and inauspicious effects, subtle karma. |
| Akshavedamsha (D-45) | D-45 | 0°40' | General indications — all matters, comprehensive reading. |
| Shashtiamsha (D-60) | D-60 | 0°30' | Karma from past lives. The finest and most karmic divisional chart. |
Deep Panchang Mathematics
Mean Motion from Mahayuga Constants
Given (Aryabhatiya):
Revolutions of Moon in Mahayuga = 57,753,336
Sidereal year length = 365.25875... days
Days in Mahayuga = 4,320,000 × 365.25875... = 1,577,917,500 days
Mean daily motion of Moon:
n☽ = 57,753,336 × 360° / 1,577,917,500 days
= 13.17635°/day
Modern: 13.17640°/day (error: 0.0004%)
Synodic month:
Tsyn = 360° / (n☽ − n☉)
= 360° / (13.17635° − 0.98563°/day)
= 29.5307 days
Modern: 29.5306 days (error: 0.0003%)
Eclipse Prediction — Iteration Method
Half-duration of eclipse (Aryabhatiya):
s = √[(r₁ + r₂)² − β²]
r₁ = sum of apparent radii (shadow + Moon)
β = lunar latitude at conjunction
Corrected duration with parallax:
λ = Lambana (east-west parallax in time)
ν = Nati (north-south parallax in latitude)
β_true = β − ν (corrected latitude)
t_first_contact = conjunction_time − s/vrel + λ
Iterative correction (Brahmagupta):
Iterate 2–3 times until t converges to <1 second
Brahmagupta added this over Aryabhata
because Moon's speed varies near eclipse
Ahargana — Day Count from Epoch
Epoch: Kali Yuga start = Feb 18, 3102 BCE
Ahargana = days elapsed since epoch
Aryabhatiya formula:
y = Kali year (e.g., 5127 for 2026 CE)
A = floor(y × 365.25875) + day_of_year
A(2026) ≈ 1,872,756 days
Planet position:
θ = (revolutions × A / mahayuga_days) × 360°
θ mod 360° = current mean longitude
Yoga Calculation (5th Panchang Limb)
Yoga = (θSun + θMoon) / 13°20'
27 Yogas (like 27 Nakshatras), but combined
Example: Sun at 45° + Moon at 130° = 175°
175° / 13.333° = 13.12 → Yoga 13 = Vyatipata
The 27 Yogas span types:
1–3: Vishkambha–Saubhagya (mixed)
6: Atiganda (very difficult)
17: Vyatipata (highly inauspicious)
18: Variyan (auspicious)
27: Vaidhriti (avoid all new work)
Karana — Half-Tithi Mathematics
Karana = (θMoon − θSun) / 6°
11 types: 4 fixed (Sakuni, Chatushpada, Naga, Kimstughna)
7 movable (Bava, Balava, Kaulava, Taitila, Garaja, Vanija, Vishti)
60 Karanas per lunar month (30 tithis × 2 half-tithis)
The 4 fixed Karanas appear once per month at specific positions
Why Karanas Matter:
Vishti Karana (Bhadra) — the most inauspicious half-tithi. No auspicious work should begin during Vishti. Ancient maritime trade routes avoided departures during Vishti — modern port activity analysis (pre-18th century Indian ports) shows statistically significant reduction in departure records during these windows.
Chakravala — The Cyclic Method for Indeterminate Equations
Brahmagupta (628 CE) and Bhaskara II (1150 CE) developed the Chakravala (चक्रवाल — “cyclic”) method for solving Nx² + 1 = y² — the Pell equation. This problem was not solved in Europe until Lagrange in 1768 — over 1,100 years later. The algorithm is a precursor to modern continued fractions and the NUCOMP algorithm in computational number theory.
The Chakravala Algorithm
Solve: Nx² + 1 = y²
Step 1: Start with a “samasa” (auxiliary equation):
Nx² + k = y² for some k (often k = ±1, ±2, ±4)
Step 2: Find m such that (a + m)² ≡ N (mod k)
and |m² − N| / |k| is minimized
Step 3: New solution (x', y', k'):
x' = (ax + y) / |k|
y' = (Nax + my) / |k|
k' = (m² − N) / k
Repeat until k = 1 → solution found
Example: N = 61
61x² + 1 = y²
Euler failed on this in 1732
Bhaskara II solved it via Chakravala:
x = 226,153,980
y = 1,766,319,049
Verify: 61 × 226,153,980² + 1 = 1,766,319,049² ✓
Madhava's Infinite Series (c. 1350–1425 CE)
Madhava of Sangamagrama — the founder of the Kerala school of mathematics — derived infinite series for π and trigonometric functions 250 years before Newton and Leibniz.
Madhava–Leibniz series for π:
π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − ...
= Σ (−1)ⁿ / (2n+1), n = 0 to ∞
Madhava's arc-sine series:
arcsin(x) = x + x³/(2·3) + (1·3)x⁵/(2·4·5) + ...
Madhava's correction term (c. 1380):
π ≈ 4[1 − 1/3 + ... ± 1/n ∓ 1/(n²+1)]
This correction term reduces error by factor of ~n²
Gregory–Leibniz rediscovered this in 1671–1674
Katapayadi — Mathematics as Poetry
All these numbers were preserved as Sanskrit verses using the Katapayadi system — a encoding where consonants map to digits (0–9). Mathematical formulas were embedded in devotional verses, ensuring they survived unchanged for millennia because religious texts were memorized with extreme precision.
ka=1, kha=2, ga=3, gha=4, nga=5
ca=6, cha=7, ja=8, jha=9, nya=0
“anūnānānanunnanānananānunnanānananunna”
→ 3.14159265358979324 (17 digits of π)