The formula
ascendant = atan2(cos LST, −(sin LST · cos ε + tan φ · sin ε))
Where it comes from
The sun sign comes from the date and is a lookup table, but the ascendant is not: it is the point of the ecliptic rising over the eastern horizon at the instant and place of birth, so it needs the time, the latitude and the longitude. It comes from the local sidereal time — which says what part of the sky is crossing the meridian — and the obliquity of the ecliptic, the tilt of the Earth orbital plane. That is positional astronomy, the same used to aim a telescope.
How to work it out by hand
- The sun sign comes straight from the date: twelve stretches of the year
- Convert the date and time to a Julian day
- Work out sidereal time at Greenwich and add the longitude of the place
- From the sidereal time, the latitude and the obliquity comes the rising degree
What is worth knowing
The ascendant advances one degree every four minutes, so it changes sign roughly every two hours: a birth time noted wrongly gives a different ascendant, which is why charts built on an approximate time are useless for this. The time has to be universal, not local: born at eight in the morning in Spain in summer means six universal. And above the polar circles the ascendant stops being defined for part of the year, so it is not calculated there.