Vaṭeśvara was an early Indian astronomer.
He wrote the Template:Ref-VSi. In it, he gives a long list of calculations in vague terms, stating the "givens" and what should be able to be derived from the given. With each calculation, he indicates that only a person who can do it should be considered an astronomer. 
- Finding the ahargaṇa, from the solar days elapsed.
- The ahargaṇa since the beginnings of various yugas, from the 'abraded multipliers and divisors'
- The mean longitude of a planet, from the ahargaṇa
- The lunar days, from the sidereal days
- Longitude of Sun and Moon without using ahargaṇa
- Sun's longitude from the Moon's
- Sun's longitude from the planets longitude, and vice versa
- Planet longitude from ahargaṇa and revolutions of another planet
- Planet longitude from 'omitted days'
- Planet longitude from some equations involving other planets
- Lord of the weekday from the 7th weekday
- Retrograde longitude of the planets
- Number of asterisms revolutions in a yuga, from the rising planet times in a yuga
- Degrees of bhuja, from degrees of koṭi, and vice versa
- kendra from the bhuja
- mean planet from planet's kendra
- jyā of bhuja, from degrees of bhuja, and vice versa
- koṭijyā from bāhujyā, and vice versa
- utkrama-jyā from jyā, and vice versa
- conversions from mean planet to true planet and back
- omitted days
- true planet from ahargaṇa
- calculate bhujajyā and koṭijyā without a jyā table
- kendra and phala
- sun crossing of nakṣatras
- The vyatipāta and the vaidhṛta
- bhogas of the nakṣatras
- many, many, many other calculations.
- Knowing the 8 types of planet motion (see below)
Vaṭeśvara described eight types of planetary motion: in the normal direction he broke it down into very fast, fast, mean, slow, and very slow. Due to the geocentric model, planets also sometimes appeared to move 'backwards', and so he also described retrograde, very retrograde, and re-retrograde motion. 
He described divisions of time as follows.
|pierce a lotus flower||truṭi|
|4.5 nimeṣa||'long syllable'|
|4 long syllables||kāṣṭhā|
|1.5 kāṣṭhā||asu, respiration (~4 seconds)|
|6 asu||sidereal pala / caṣaka / vināḍi / vighaṭikū|
|60 sidereal pala||ghaṭikā (~24 minutes)|
|43,20,000 solar year||yuga|
|2 kalpa||day and night of Brahmā|
|100 Brahmā years||life of Brahmā / Mahākalpa|
- 1 & 2: using two shadows of a śaṅku (gnomon), one in morning and evening
- 3: When the sun is in the prime vertical
- 4: a different use of śaṅku shadow
- 5: the rising and setting points of any cosmic object
- 6: The rising point of Revatī or Śravaṇa, or the midpoint of Citra and Svāti (asterisms).
- 7: yet another use of shadow, this time three shadows. Also using fish figures
- 8: When the sun is at Aries or Libra, the equinoctial midday-shadow is South/North