PROMINENT ANCIENT INDIAN ASTRONOMERS

1> Acharya Lagadha Vedang Jyotish 1500 BC

2> Aryabhat Aryabhatiya 500 AD

3> Varahamihir Panchsiddhantika 505 AD

4> Bhaskara 1 Mahabhaskriya 600 AD

5> Lallacharya Shishadhivridhidatantra 638 AD

6> Brahmagupta Brahmasphutsiddhanta 628 AD

7> Vateshwar Vateshwarsiddhanta 880 AD

8> Munjala Laghumanasam 932 AD

9> Aryabhat 2 Mahasiddhanta 950 AD

10> Bhaskaracharya 2 Siddhantashiromani 1150 AD

11> Ganesh Daivadna Grahalaghav 1520 AD

PaaOilaSa raomak vaisaYz saaOr pOtamahastu pMcaisaQdaMta:

AaQauinak saUya-isaQdaMt i#a`stabd 1000

PROMINANT ASTRONOMERS FROM KERALA

1> Madhavan of Sangamagram (1340-1425) Karanapaddhati(1375)

2> Narayana Pandit (1340-1400) Ganita Kaumudi

3> Parameshvara (1370-1460) Drigganita

4> Nilakantha Somayaji (1444-1544) Tantrasangraha

5> Jyesthadeva (1500-1610) Yuktibhasa

6> Sankara Variar (1500-1560) Kriya-kramakari

7> Sankar Varman (1774-1839) Sadratnamala

CONCEPT OF YUGAS

Kali 432000 Years

Dwapar 864000 Years

Treta 129600 Years

Krita 1728000 Years

Mahayuga = Kali + Dwapar + Treta + Krita

= 4320000 Years

Manavantar = 71 Mahayugas + 1 Krita

= 71 x 4320000 + 1728000

= 308448000 Years

Kalpa = 14 Manvantaras + 1 Krita

= 14 x 308448000 + 1728000

= 4320000000 Years

Kali Yuga started on 17/18 Feb 3102 BC

CONCEPT OF NAKSHTRAS

Period of revolution of Moon = 27 Days

Total angle covered in this period = 360 degrees

Speed of revolution per day = 360/27 = 13.33 deg/day

One Nakshtra = 13.33 degrees

There are 27 Nakshtras

Moon crosses one Nakshtra per day

Names of 27 Nakshtras

1> Ashvini 10> Magha 19> Mula

2> Bharani 11> Purva Phalguni 20> Purvashadha

3> Krittaka 12> Uttara Phalguni 21> Uttarashadha

4> Rohini 13> Hasta 22> Shravana

5> Mrigashirsha 14> Chitra 23> Dhanishta

6> Aardra 15> Swati 24> Shatataraka

7> Punarvasu 16> Vishakha 25> Purvabhadrapada

8> Pushya 17> Anuradha 26> Uttarabhadrapada

9> Ashlesha 18> Jyeshta 27> Revati

tO%trIya saMihtotIla maihnyaaMcaI naavao

maQau maaQava vasaMtSau@r Sauica g`aIYmanaBasa\ naBasya vaYaa-[Ya }ja- Sardsahs\a\ sahsya homaMttpsa\ tpsya iSaiSar

BaartIya maihnao1Å caO~ 7Å AiSvana2Å vaOSaaK 8Å kait-k3Å jyaoYz 9Å maaga-SaIYa-4Å AaYaaZ 10Å PaaOYa5Å EaavaNa 11Å maaGa6Å Baad`pd 12Å falgauna

caMd` paOiNa-maolaa jyaa naxa~acyaa javaL Asatao %yaa naxa~acao naava maihnyaalaa doNyaat Aalao.PaaOiNa-maolaa ra~Bar to to naxa~ idsato.

caMd` naxa~maasa duNaa itqaI gaNaa tIna ]Naa naxa~ jaaNaa

Aa.sa.kbaro

SIGNS OF ZODIAC

1> MESH ARIES 14 APRIL

2> VRISHABHA TAURUS 14 MAY

3> MITHUN GEMINI 14 JUNE

4> KARKA CANCER 14 JULY

5> SIMHA LEO 14 AUGUST

6> KANYA VIRGO 14 SEPTEMBER

7> TULA LIBRA 14 OCTOBER

8> VRISHIKA SCORPIUS 14 NOVEMBER

9> DHANU SAGITTARIUS 14 DECEMBER

10> MAKAR CAPRICORNUS 14 JANUARY

11> KUMBHA AQUARIUS 14 FEBRUARY

12> MEENA PICES 14 MARCH

INDIAN WESTERN ENTRY OF SUN

DECLINATION OF THE SUN THROUGHOUT AN YEAR

itqaI

Angular speed of Sun = 1 degree per day

Angular speed of Moon = 13.33 degrees per day

Angular speed of Moon with respect Sun = 13.33 – 1

= 12 deg/day

One Tithi = 12 degrees nearly

Tithi is also angular difference between Sun and Moon

Angular difference in white fortnight = 12 x Tithi

Let Tithi = 4, Angular difference = 4 x 12 = 48 degrees

Angular difference in black fortnight = 180 – 12 x Tithi

Let Tithi = 3, Angular difference = 180 – 12 x 3 = 144 degrees

Tithi also indicates the Phase of the Moon

From the Tithi, time of Moon rise can also be estimated.

For example, in the third Tithi of dark fortnight Moon rise

Will be about 2.5 ( 36/15) hours after Sunset.

In the fifth Tithi of white fortnight Moonrise

Will be about 4 (60/15) after Sunrise.

From the Tithi time of high tide can also be estimated.

itqaI vaRiQd AaiNa xaya • At the time of sunrise the current tirhi is the tithi of the day

• The tithi which exists at the consecutive sunrise is the case of

tithi vriddhi or interval of tithi > 24 hours.

• The tithi which falls between the two consecutive sunrises is the

case of tithi kshya or interval of tithi < 24 hours

Tithi Cycle ( Metonic Cycle)

• 19 tropical years = 19 x 365.2422 = 6939.602 days

• 235 synodic months of Moon = 235 x 29.3 = 6939.689 days

• Hence after 19 years sequence of tithis are repeated

A difference of 1 day after 218.5 years.

INTERCALARY MONTH (AiQak maasa )

• Lunar month is an interval between two successive new moons.

Which is equal to 29.530589 days.

• one lunar year = 29.530589 x 12 = 354.36706 days

• one solar year = 365.25636 days.

• Difference between solar and lunar year = 10.8893 days.

• If the difference is allowed to accumulate, Lunar months

will be out of steps with the seasons. Hence idea of intercalary

month was introduced.

• When the difference adds up to a full lunar month, intercalary

month is added in that particular lunar month.

• Intercalary month occurs after 10.8893 x 3 = 32.6679 months,

or after about 33 months.

• Normal lunar year consists of 12 lunar months of 354 0r 355 days.

In a lunar year in which there is intercalary month will have 13

lunar months of 383 or 384 days.

AiQak maasa

xaya maasa

FORTNIGHTS AND ELONGATION

(pxa AaiNa [naaMtr)• Two Fortnights White (Sau@la) and Dark (kRYNa)• In White fortnight angular distance between Sun and Moon increases.

• In Dark fortnight angular distance between Sun and Moon decreases.

• In White fortnight Moon rises after sunrise and sets after sunset.

• In Dark fortnight Moon rises before sunrise and sets before sunset.

• Elongation is angular difference between longitudes of Sun and Planet/star.

• If longitude of a star or planet is greater than longitude of Sun,

elongation is east.

• If longitude of a star or planet is less than longitude of Sun,

elongation is west.

• If elongation is east star or planet will rise after sunrise.

• If elongation is west star or planet will rise before sunrise.

• Time of rise of a star or planet = elongation/15.

KNOWING THE NAME OF THE MONTH, THE TYPE OF THE FORTNIGHT

AND THE TITHI, ENTIRE SKY IS KNOWN.

Angle between the Sun, Earth and Planet is called Elongation

Longitude of Sun > Longitude of planet : Elongation is west

Elongation = Longitude of Sun – Longitude of planet

If Elongation is West then planet rises before sunrise and sets before

sunset

Longitude of planet > Longitude of Sun : Elongation is east

Elongation = Longitude of planet – Longitude of Sun

If elongation is East then planet rises after sunrise and sets after sunset

ELONGATION

elongation2

]<arayaNa AaiNa dixaNaayana• After Winter Solstice uttarayan starts.

• At present Uttarayan starts on 22 December every year

• After Summer Solstice Dakshinayan starts

• At present Dakshinayan starts on 22 June every year.

• After onset of Uttarayan day length increases night length decreases.

• After onset of Dakshinayan day length decreases night length increases.

• Lengths of day and night are equal on Vernal Eqinox (vasaMtsaMpat) and

Autumnal Equinox (SardsaMpat).longitude

22 Dec

22 JunSummer Solstice

Winter Solstice

Niryan

AYANANSHA

(ica~a)

vaodaMga jyaaoitYayaqaa iSa#aa mayauraNaama\ naagaanaama\ maNayaao yaqaatqaOva sava- Saas~aNaama\ jyaaoitYama\ maUQa-ina itYzit

Aacaaya- lagaQa

EVENT JULIAN GREGORIAN

WINTER SOLSTICE 03 JAN 1752BC 19 DEC 1753 BCFUL MOON DAY 17 JAN 1752 BC 02 JAN 1752VERNAL EQUINOX 04 APL 1752 BC 20 MAR 1752 BCSUMMER SOLSTICE 05 JUL 1752 BC 20 JUN 1752 BC

DIFFERENCE BETWEEN GREGORIAN AND JULIAN 15 OR 16 DAYS

GREGORIAN AND JULIAN CALENDAR DATES OF VEDANGA JYOTISH

WINTER SOLSTICE

NOW 22 DECEMBER

SUMMER SOLSTICE

NOW 22 JUNE

SUN

MOOMOON

3 JANUARY 1752 BC NEW MOON DAY IN DHANISHTA

17 JANUARY 1752 BC FULL MOON DAY IN MAGHA

4 APRIL 1752 BC VERNAL EQUNOX IN BHARANI

5 JULY 1752 BC DAKSHINAYAN IN MID OF ASHLESHA

SUN

SHIFT OF VERNAL EQUINOX

1 : Earth rotates

2 : PI = 3.1416

3 : Sidereal day = 23hrs 56 min 4.1 sec

4 : Diameter of Earth = 12810 km

5 : Earth is round

6 : length of Earth’s shadow = 1392000km

7 : Solution of first order indeterminate equation

8 : Sidereal year = 365.2563

9 : Revolution of Moon = 27.3216 days

11 : Period revolutions of the planets (days)

planet Aryabhat modern

Sun 365.25868 365.25636

Moon 27.32167 27.32166

Mercury 87.96988 87.9693

Venus 224.69814 224.7008

Mars 686.99974 686.9797

Jupiter 4332.2721 4332.5887

Saturn 10766.06465 10759.201

GLIMPSES ARYABHATIYAARYABHAT : BORN IN 476 AD

Sines in terse verse : Roddam Narasingh : Nature 20/27 DEC 2001

maiK BaiK fiK QaiK naiK HaiK =iK hsJa skik ikYga SGaik ikQvaGlaik ikga` h@ya Qaik ikca sga JaSa =va @la Pt f C klaaQa-jyaa:

arc ABC = radius R

Angle AOC = 57deg 17 min 44 sec = 3438 minutes

All reference circles will have, Radius 3438

For example: sin(3.75deg) = sin(225min) = 225/3438 = 0.0654

radian

R

R

R

saPtOto haoroSaa: SanaOScaraVa yaqaak`maM SaIGa`a:SaIGa`k`maat\ catuqaa- BavaMit saUyaao-dyaat\ idnapa:

weekdaysHOURS OF A DAY

(HORA)

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24

SATURN

SUN

SATURN

JUPITER

MARS

SUN

VENUS

MERCURY

MOON

SLOWEST

FASTEST

THE OF THE WEEK

1: caturiQakM Satma\\ AYTgauNama = 104 x 8 = 832

2: vdaYaiYTstqaa sahs~aNaama = 62000

(1) + (2) = 62000 + 832 = 62832

Ayautvdya = 20000

π = 62832/20000 = 3.146 (approximately)\

The most accurate value among those of ancients.

caturiQakM Satma\\ AYTgauNama\ vdaYaiYTstqaa sahs~aNaama\Ayautvdya ivaYkmBasya Aasanna: vaR<apirNa:

VALUE OF π

The gods living at north pole and demons living at south pole

See the Sun for six months when it rises.

The ancestors living on Moon see the Sun for half of the lunar

month i.e. for 15 days.

The people living on Earth see the Sun for half of the day i.e.

for 12 hours.

The meaning of this verse is,

On north and south pole the day and night cycle is of six months.

On Moon the day and night cycle if of 15 days.

On Earth the day and night cycle is of 12 hours.

Day and night cycle

On north and south pole, on Moon and the Earth

rivavaYaaQa-M dovaa: pSyaMit ]idtM rivMa tqaa pota:SaiSamaasaaQa-M iptr: SaiSagaa: kuidnaaQa-M [h manaujaa:

Planet Bhagana Sidereal time of revolution (days)

Aryabhata Modern

Sun 4320000 365.25868 365.25636

Moon 57753336 27.32167 27.32166

Mercury 17937020 87.96988 87.9693

Venus 7022388 224.69814 224.7008

Mars 2296824 686.99974 686.9797

Jupiter 364224 4332.27217 4332.5887

Saturn 146564 10766.06465 10759.201

Ascending node 232226 6794.74951 6793.39108

BHAGANANumber revolutions around the Earth in Mahayuga

SYNODIC TIME = NUMBER OF YEARS IN YUGA/NUMBER OF CONJUNCTIONSNUMBER OF YEARS IN YUGA = 4320000NUMBER OF CONJUNCTIONSIN YUGA = N1 – N2 N= REVOLUTIONSSYNODIC TIME OF SUN AND MOON = 4320000/(57753336 – 4320000)

= 4320000/5343336 = .0808484 X 365.25868= 29.530582 days

Planet Aryabhatiya Modern

Mercury 0.375 0.387

Venus 0.725 0.723

Mars 1.538 1.523

Jupiter 5.16 5.2

Saturn 9.41 9.54

Heliocentric distances of the planets

in terms of Earth to Sun distance

Planet Aryabhatiya(days) Modern(days)

Mercury 115.88 115.975

Venus 583.89 585.92

Mars 779.92 779.93

Jupiter 398.89 398.75

Saturn 378.08 378.00

SYNODIC TIMES OF THE PLANETS

PLANET AVERAGE DAILY SPEED(DEG)

ARYABHAT MODERN

Moon 13.176333 13.178029

Mercury 4.092029 4.0953717

Venus 1.6019858 1.6013164

Sun 0.9856377 0.9861002

Mars 0.524014 0.5234908

Jupiter 0.0830973 0.0833221

Saturn 0.0334383 0.03330

AVERAGE DAILY SPEEDS OF THE PLANETS

Epycycle

MV = Equation of Center ( maMdfla )

At A and C

Sun : 840 min

Moon : 1920 min

At B and D

Sun : 820 min

Moon ; 1900 min

]<ar QaRvalanka-p23

]jaO[ina

ivaYauvavaR<alaMka

dixaNa QaRva (vaDvaanala)

]jaO[ina laMkayaa: tt\ caturMSao samaao<art:]jaO[ina AxaaMSa 22.5 ],<ar (vaastva 23.1)

LANKA AND FOUR CARDINAL CITIES

]dyaao laMkayaama\ sa: Astmaya: saivatu: eva isaQdpuromaQyaa*nao yavakaoTyaaM raomakivaYaya: AQa-ra~: syaat\

four-cities

isaQdpUr

raomak

laMka

yavakaoiT

saUya-ikrNa

saUya-ikrNa

Brahmagupta occupies an important place in the history

of Oriental culture. Brahmagupta taught astronomy to

the Arabs before they came to know Ptolemy’s work,

since, references to the works, ‘Sindhind’ , and ‘Al-

Arkand frequently occur in Arabic literature. These are the translations of Brahmagupta’s works,

‘BrahmaSiddhanta’, and ‘Khandakhadyka’ .

Prof.Sachau, in the translation of ‘ Al-Biruni’s India

BRAHMAGUPTA (BORN IN 598 AD)

BRAHMAGUPTA (BORN IN 598 AD)

First to give the formula for the area of Cyclic Quadrilateral

In trigonometry,

the Brahmagupta interpolation formula

is a special case of the Newton-Stirling interpolation formula,

which calculates the values of sine at different intervals.

The formula was developed by Brahmagupta in 665,

which was later expanded by Newton and Stirling

around a thousand years later to develop

the more general Newton-Stirling interpolation formula.

The Brahmagupta interpolation formula is defined as:

Angle Indian sine First difference Second difference

0 0

15 39 39

30 75 36 -3

45 106 31 -5

60 130 24 -7

75 145 15 -9

90 150 5 -10

150Sine (67) = sine (60) + sine(7) = 130 + {(7/15)(15 + 24)/2 + (7/15)(15 – 24)/2}

= 130 + 8.12 = 138.12

Sine(67) = 138.12/150 = 0.9208 (=9205 actual)

Glimpses of Siddhantshiromani

1 : Idea of infinity : Khahara rashi

2 : Numbers from one to 10^18

3 : PI = 3.1416

4 : Idea of calculus d(sin x) = cos x dx

5 : Khandameru = Pascal trtiangle

6 : Two line proof of so called

Pythagoras hypothesis

7 : General solution of Indeterminate

equation of second order

61x^2 + 1 = Y^2

X=226153980 Y = 1766319049

8 : Circumference of Earth = 40000km

SIDDHANTASHIROMANI

1 : Lilavati

2 : Beejaganit

3 : Ganiytadhyaya

4 : Goladhyaya

BHASKARACHARYA (1114 -1183 AD)

BIRTH AND DATE OF WRITING SIDDHANTASHIROMANI

RASA = 6 GUNA = 3 PURNA = 0 MAHI = 1

BHASKARACHARYA WAS BORN IN SHAKE 1O36

THAT IS 1036 + 78 = 1114 AD

SIDDHANTASHIROMANI WAS WRITTEN AT THE AGE

RASA = 6 GUNA = 3 THAT IS

WHEN HE WAS 36 YEARS OLD

VALUE OF PI

bha = 27 nanda = 9 agni = 3 kha = 0 bana = 5 suryai = 12

PI = 3927/1250 = 3.1416

PI = 22/7

OR

EDUCATION OF BHASKARACHRYA

NAMES OF THE NUMBERS

THE PLACE AND THE TEACHER

BHASKARACHARYA

PERIODIC TIME OF THE PLANETS

Planet Revolutions in Kalpa Periodic Time (years)

Sun 4,320,000,000 1

Moon 57,753,300,000 27.3 days

Mercury 17,936,998,984 87.96 days

Venus 7,022,389,492 224.76 days

Mars 2,296,828,522 1.88

Jupiter 365,226,455 11.86

Saturn 146,567,298 29.47

Moon’s Node 2,323,111,698 18.56

Moon’s Perigee 488,105,858 8.85

TIME UNITES

1 Muhurta = 24/30 hours = 48 minutes

1 Ghatika = ½ Muhurta = 24 minutes

1 Kala = 1/30 Ghatika = 48 seconds

1 Kashtha = 1/30 Kala = 8/5 seconds

1 Nimesh = 1/18 Kashtha = 4/45 seconds

1 Tatpar = 1/30 Nimesh = 2/675 seconds

1 Truti = 1/100 Tatpar = 2/67500 seconds

Aqa kismaMiScat puro AxaaMSaana\ &a%vaa tsmaat\ purat\ ]<art: Anyaismana\ puro &oyaa:tt: toYaama\ AMtraMSaO: purantr yaaojanaO: ca Anaupat:yat\ AMtraSaO: puraMtr yaaojanaaina laByanto tda cak`aMSaO: ikma\ [itflaM BaUPairiGa yaaojanaaina

lunar eclipse

D

D1

L

Length of the Earth’s shadow = L = (2Re x D)/(2Rs – 2Re) = 1392000 km

Re = radius of Earth Rs = radius of Sun D = Earth - Sun distance

L = (Earth’s diameter x Earth-Sun distance)/(Sun’s diameter-Earth’s diameter)

Diameter of Earth’s shadow at Moon’s distance=d = (L -D1)2Re/L = 9000 km

D1 = Earth - Moon distance 2Re = diameter of Earth

caMd`ga`hNa

MADHAV OF SANGAMGRAMAM (1350 – 1435)

Greatest of the Indian medieval Astronomer- Mathematician

POWER SERIES

Madhav Gregory Series

Madhav Newton Series

Series for PI

ga`hlaaGava

• Written by Ganesh Daivadna in 1520 AD.

• Accepted 11 year cycle to reduce number of days from the epoch.

• 11 year cycle consists of only 4016 days.

• Removed the use of trigonometry. Still calculations are correct.

• Simplified the calculations.

• Accepted all over India form almanac preparations.

• Epoch Chaitra Shukla Pratipada Shake 1442 (1520 AD)

• Ganesh wrote more than 15 Astronomical books.

• Most brilliant Astronomer of Medieval India.

jyaacaapkma-rihtM saulaGaup`karma\ ktR-ma\ ga`hp`krNaM sfuTma\ ]Vtaosisma

ganesh2

gaNaoSa gauNaakar

JANTARMANTAR JAIPUR

SAVAI JAISINGH

(1686-1743)

SAMRAT YANTRA

SAMRAT YANTRA

ASTRONOMICAL INSTRUMENTS

Instrument Description Use

Chap One half of the Protractor Altitude of celestial body

Turiya One quarter of Protractor Altitude of celestial body

Nadiwalaya Two fixed circular plates Time and meridian passage

parallel to equator of Sun

and Gnomon

Shanku Vertical Sundial N-S line, Latitude,Time

Yashti Straight staff Height, Altitude, Latitude

Chakra A ring mounted between Declination, Hour Angle

two posts along N -S line

Jayprakash A hollow hemisphere with Coordinates of celestial body

number of inscribed arcs

Ram A huge cylinder with its Altitude, Azimuth

height equal to its radius

Bhitti N – S wall with circular Meridian passage of Sun, stars

marking and Gnomon

Samrat Massive Gnomon with Accurate time

equatorial half circles

marked in degrees and minutes

Ghatika A hollow bowl with hole at bottom Time

shanku yantra copy.jpg

shanku yantra2 copy.jpgNORTH SOUTH DIRECTION N

S

shadow 21 march copy.jpg

MARCHSHADOW21

Sin (longitude ) = sin (declination)/sin(obliquity of ecliptic)

22 JUNE SHADOW

DECLINATION MARKINGS