Āryabhaṭīya/Chapter 01
प्रणिपत्य एकं अनेकं कं सत्यां देवतां परं ब्रह्म ।
आर्यभटस् त्रीणि गदति गणितं कालक्रियां गोलम् ॥ १.१ ॥
Having paid obeisance to God Brahma — who is one and many, the real God, the Supreme Brahman - Aryabhata sets forth the three, viz., mathematics (ganita), reckoning of time (Kalakriya) and celestial sphere (gola).
Āryabhaṭīya/Chapter 01
वर्गाक्षराणि वर्गे अवर्गे अवर्गाक्षराणि कात् ङ्मौ यस् ।
खद्विनवके स्वरास् नव वर्गे अवर्गे नव अन्त्यवर्गे वा ॥ १.२ ॥
The varga letters (k to m) (should be written) in the varga and the avarga letters (y to h) in the avarga places. (The varga letters take the numerical values 1, 2, 3, etc.) from k onwards; (the numerical value of the initial avarga letter) y is equal to n plus m (i.e., 5 + 25). In the places of the two nines of zeros (which are written to denote the notational places), the nine vowels should be written (one vowel in each pair of the varga and ayarga places). In the varga (and avarga) places beyond (the places denoted by) the nine vowels too (assumed vowels or other symbols should be written, if necessary).
In this system, consonants represent numbers and vowels represent place values. Aryabhata used the 33 consonants of Sanskrit to encode digits, and 9 vowels to indicate powers of ten — thereby creating a true place value notation centuries before modern numerals.
1. Consonants and their numeric values
Varga letters (क to म) — base digits 1 to 25:
क = 1 ख = 2 ग = 3 घ = 4 ङ = 5
च = 6 छ = 7 ज = 8 झ = 9 ञ = 10
ट = 11 ठ = 12 ड = 13 ढ = 14 ण = 15
त = 16 थ = 17 द = 18 ध = 19 न = 20
प = 21 फ = 22 ब = 23 भ = 24 म = 25
Avarga letters (य to ह) — used for higher groups, beginning after म (25):
य = 30 र = 40 ल = 50 व = 60 श = 70 ष = 80 स = 90 ह = 100
2. Vowels and their place values
Each vowel marks a place value, analogous to powers of ten.
अ = 10⁰ (units)
इ = 10¹ (tens)
उ = 10² (hundreds)
ऋ = 10³ (thousands)
ऌ = 10⁴ (ten-thousands)
ए = 10⁵ (lakhs)
ऐ = 10⁶ (ten-lakhs)
ओ = 10⁷ (crores)
औ = 10⁸ (ten-crores)
3. Writing numbers
A consonant gives the digit, and the vowel attached to it gives its positional value.
For example:
क + अ = 1 × 10⁰ = 1
क + इ = 1 × 10¹ = 10
ग + उ = 3 × 10² = 300
न + ओ = 20 × 10⁷ = 2 × 10⁸ = 200,000,000
4. Summary of rules
(1) Varga letters (क–म) represent numbers 1–25.
(2) Avarga letters (य–ह) represent larger values (30, 40, … 100).
(3) Vowels mark the positional place (10⁰, 10¹, … 10⁸).
(4) Writing a consonant with a vowel encodes a digit × place.
(5) For places beyond nine vowels, vowels or other markers are reused cyclically.
(6) This system, unlike earlier additive notations, is positional — the same consonant has different value depending on the vowel attached.
Āryabhaṭīya/Chapter 01
युगरविभगणास् ख्युघृ शशि चयगियिङुशुछ्लृ कु ङिशिबुण्ल्ष्खृ प्राक् ।
शनि ढुङ्विघ्व गुरु ख्रिच्युभ कुज भद्लिझ्नुखृ भृगुबुधसौरास् ॥ १.३ ॥
In a yuga, the eastward revolutions of the Sun are 43,20,000; of the Moon, 5,77,53,336; of the Earth, 1,58,22,37,500; of Saturn, 1,46,564; of Jupiter, 3,64,224; of Mars, 22,96,824; of Mercury and Venus, the same as those of the Sun
Āryabhaṭīya/Chapter 01
चन्द्रौच्च र्जुष्खिध बुध सुगुशिथृन भृगु जषबिखुछृ शेष अर्कास् ।
बुफिनच पातविलोमास्बुधाह्नि अजार्कौदयात्च लङ्कायाम् ॥ १.४ ॥
of the Moon's apogee, 4,88,219; of (the sighrocca of) Mercury, 1,79,37,020; of (the Sighrocca of Venus, 70,22,388; of (the Sighroccas of) the other planets, the same as those of the Sun; of the moon's ascending node in the opposite direction (i.e., westward), 2,32,226.4 These revolutions commenced at the beginning of the sign Aries on Wednesday at sunrise at Lanka (when it as the commencement of the current yuga).
Āryabhaṭīya/Chapter 01
काहस्मनवस् ढ मनुयुगास् श्ख गतास्ते च मनुयुगास् छ्ना च ।
कल्पऽदेस् युगपादास् ग च गुरुदिवसात्च भारतात्पूर्वम् ॥ १.५ ॥
A day of Brahma (or a Kalpa) is equal to (a period of) 14 Manus, and (the period of one) Manu is equal to 72 yugas. Since Thursday, the beginning of the current Kalpa, 6 Manus, 27 yugas and 3 quarter yugas had elapsed before the beginning of the current Kaliyuga (lit. before Bharata).
Āryabhaṭīya/Chapter 01
शशिराशयस् ठ चक्रं ते अंशकलायोजनानि यवञगुणास् ।
प्राणेन एति कलां भं खयुगांशे ग्रहजवस्भ वांशे अर्कस् ॥ १.६ ॥
Reduce the Moon's revolutions (in a yuga) to signs, multiplying them by 12 (lit. using the fact that there are 12 signs in a circle or revolution). hose signs mutiplied successively by 30, 60 and 10 yield degrees, minutes and yojanas, respectively. (These yojanas give the length of the circumference of the sky). The Earth rotates through (an angle of) one minute of arc in one respiration (=4 sidereal seconds). The circumference of the sky divided by the revolutions of a planet in a yuga gives (the length of) the orbit on which the planet moves. The orbit of the asterisms divided by 60 gives the orbit of the Sun.
Āryabhaṭīya/Chapter 01
नृ षि योजनम् ञिला भूव्याससर्कैन्द्वोर् घ्रिञा गिण क मेरोस् ।
भृगुगुरुबुधशनिभौमास्शशि ङ ञ ण न मांशकास्समार्कसमास् ॥ १.७ ॥
8000 nr make a yojana. The diameter of the Earth is 1050 yojanas; of the Sun and the Moon, 4410 and 315 yojanas, (respectively); of Meru, 1 yojana; of Venus, Jupiter, Mercury, Saturn and Mars (at the Moon's mean distance), one-fifth, one-tenth, one-fifteenth, one-twentieth, and one-twentyfifth, (respectively), of the Moon's diameter. The years (used in this work) are solar years.
Āryabhaṭīya/Chapter 01
भापक्रमस्ग्रहांशास्शशिविक्षेपसपमण्डलात् झ अर्धम् ।
शनिगुरुकुज ख क ग अर्धं भृगुबुध ख स्चाङ्गुलस् घहस्तस् ना ॥ १.८ ॥
The greatest declination of the Sun is 24°. The greatest celestial latitude (Jit. deviation from the ecliptic) of the Moon is 4.5°; of Saturn, Jupiter and Mars, 2°, 1° and 13° respectively; and of Mercury and Venus (each), 2°. 96 angulas or 4 cubits make a nr.
Āryabhaṭīya/Chapter 01
बुधभृगुकुजगुरुशनि न व रा ष ह गत्वा अंशकान् प्रथमपातास् ।
सवितुरमीषां च तथा द्वा ञखि सा ह्दा ह्ल्य खिच्य मन्दौच्चम् ॥ १.९ ॥
The ascending nodes of Mercury, Venus, Mars, Jupiter and Saturn having moved to 20°, 60°, 40°, 80° and 100° respectively (from the beginning of the sign Aries) (occupy those positions); and the apogees of the Sun and the same planets (viz., Mercury, Venus, Mars, Jupiter and Saturn) having moved to 78°, 210°, 90°, 118°, 180° and 236° respectively (from the eginning of the sign Aries) (occupy those positions).
Āryabhaṭīya/Chapter 01
झ अर्धानि मन्दवृत्तं शशिनस् छ ग छ घ ढ छ झ यथा उक्तेभ्यस् ।
झा ग्ड ग्ला र्ध द्ड तथा शनिगुरुकुजभृगुबुधौच्चशीघ्रेभ्यस् ॥ १.१० ॥
The manda epicycles of the Moon, the Sun, Mercury, Venus, Mars, Jupiter and Saturn (in the first and third anomalistic quadrants) are, respectively, 7, 3, 7, 4, 14, 7 and 9 (degrees) each multiplied by 4.5 (i.e., 31.5, 13.5, 31.5, 18, 63, 31.5 and 40.5 degrees, respectively); the sighra epicycles of Saturn, Jupiter, Mars, Venus and Mercury (in the first and third anomalistic quadrants) are, respectively, 9, 16, 53, 59 and 31 (degrees) each multiplied by 4.5 (i.e, 40.5, 72, 238.5, 265.5 and 139.5 degrees, respectively).
Āryabhaṭīya/Chapter 01
मन्दात् ङ ख द ज डा वक्रिणां द्वितीये पदे चतुर्थे च ।
जा ण क्ल छ्ल झ्न उच्चात्शीघ्रात् गियिङश कुवायुकक्ष्याअन्त्या ॥ १.११ ॥
The manda epicycles of the retrograding planets (viz., Mercury, Venus, Mars, Jupiter and Saturn) in the second and fourth anomalistic quadrants are, respectively, 5, 2, 18, 8 and 13 (degrees) each multiplied by 4.5 (i.e., 22.5, 9, 81, 36 and 58.5 degrees, respectively); and the Sighra epicycles of Satura, Jupiter, Mars, Venus, and Mercury (in the second and fourth anomalistic quadrants) are, respectively, 8, 15, 51, 57 and 29 (degrees) each multiplied by 4.5 (i.e., 36, 67.5, 229.5, 256.5 and 130.5 degrees, respectively). 3375 is the outermost circum- ference of the terrestrial wind.
Āryabhaṭīya/Chapter 01
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व ।
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला अर्धज्यास् ॥ १.१२ ॥
228, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, and 7— these are the Rsine.differences (at intervals of 225 minutes of arc) in terms of minutes of arc.
Āryabhaṭīya/Chapter 01
दशगीतिकसूत्रं इदं भूग्रहचरितं भपञ्जरे ज्ञात्वा ।
ग्रहभगणपरिभ्रमणं स याति भित्त्वा परं ब्रह्म ॥ १.१३ ॥
Knowing this Dasagitika-sutra, (giving) the motion of the Earth and the planets, on the Celestial Sphere (Sphere of asterisms or Bhagola), one attains the Supreme Brahman after piercing through the orbits of the planets and stars.