Little Almanac

The angles

The four angles are the answer to a question of geometry: given a moment and a place on the Earth, where does the ecliptic cross the two great circles that a standing observer’s own position defines? Everything a tradition later built on them rests on that crossing, and the crossing itself is computable to the arcsecond.

Two circles, not one axis

The distinction that most descriptions of a chart lose is that the four points come from two different circles.

The horizon is the plane through the observer separating the visible sky from the hidden. Where the ecliptic rises through it in the east is the Ascendant; where it descends in the west is the Descendant.

The meridian is the plane through the observer, the zenith and the celestial poles. Where the ecliptic crosses it above is the Midheaven, or medium coeli; where it crosses below is the imum coeli.

Two circles, two crossings, four points, and the four fall short of two perpendicular axes laid over the zodiac. The Ascendant and the Midheaven are separated by roughly ninety degrees of ecliptic longitude and almost never exactly ninety, because the angle at which the ecliptic meets the horizon changes continuously with the latitude, the season and the hour. At high latitudes the separation distorts far enough to break the systems built on it, which is why quadrant house division fails there and says so.

Correction

Commonly saidThe Midheaven is the cusp of the tenth house.

It is in the quadrant systems, because those systems define it to be: they divide the space between the Ascendant and the Midheaven and take the two angles as cusps by construction. It is not in whole sign houses, the oldest attested division, where the tenth house is the tenth sign from the rising one and the Midheaven falls wherever it falls, which may be in the ninth, the tenth or the eleventh. In whole sign the Midheaven becomes a moving point of emphasis that can land in different places on different charts, which is more interesting than the tidy version. See House Division.

Ptolemy’s own difficulty was the birth time

The sensitivity of the Ascendant to the recorded time was noticed in antiquity. It opens the third book of the Tetrabiblos, where Ptolemy names it as the first obstacle to the whole enterprise 1 bk. III ch. 2:

Difficulty often arises with regard to the first and most important fact, that is, the fraction of the hour of the birth; for in general only observation by means of horoscopic astrolabes at the time of birth can for scientific observers give the minute of the hour, while practically all other horoscopic instruments on which the majority of the more careful practitioners rely are frequently capable of error, the solar instruments by the occasional shifting of their positions or of their gnomons, and the water clocks by stoppages and irregularities in the flow of the water from different causes and by mere chance.

The same chapter proposes a repair, which shows how seriously the problem was taken. Ptolemy directs the practitioner to the new or full moon most recently preceding the birth, to establish which planet rules that degree by the five forms of domination, and then to take the degree that planet occupies as the degree rising at the nativity. Whatever else that procedure is, it is an attempt to recover an unmeasurable quantity from a measurable one.

The difficulty has moved: modern birth records give a time rounded to the minute at best and often to the quarter hour, and the rounding is now the dominant error.

How fast the angles move

The Sun, Moon and planets move slowly against the zodiac. The angles do not. They are driven by the Earth’s rotation, so the whole set turns through the zodiac roughly once a day, and the Ascendant advances at an average of about fifteen degrees an hour.

The average conceals a wide range. The rate is not constant, because the ecliptic meets the horizon at a different angle at every hour and every latitude. Signs that rise steeply take much longer to clear the horizon than signs that rise shallowly, a distinction the tradition names as signs of long and short ascension. The consequence for anyone reading a chart is that an hour of uncertainty in the birth time is not a fixed amount of uncertainty in the Ascendant.

Established

A chart’s four angles are the only values on it that depend on the place of birth. Every planetary longitude in a geocentric chart would be the same for a birth at the same instant anywhere on Earth. Move the birthplace and the planets do not move at all; the angles swing through the whole zodiac.

Where the angles stop existing

Software usually hides this part; this site’s implementation reports it. Three failure modes are real. Two are properties of the sky and the third is a property of the arithmetic.

Beyond about 89.9 degrees of latitude the horizon has closed onto the celestial equator, and the ordinary notion of a degree rising in the east no longer applies: there is no east. This site’s implementation refuses the computation at that point.

Near the latitude where the north ecliptic pole passes through the zenith, close to ninety degrees minus the obliquity, the ecliptic comes to lie along the horizon. Every degree of it rises at once. There is no single rising degree, and in the moments either side of that configuration the Ascendant slews through the zodiac at up to about a hundred and seventy-three degrees per minute of clock time. That figure is this site’s own computation from its own implementation, and it is the reason the chart tool reports an unstable Ascendant as unstable.

Gap in the evidence

The third failure mode is a fact about arithmetic, recorded here because it is the kind of error that survives review. The standard closed-form expression for the Ascendant contains a tangent of the latitude. At the pole that tangent is mathematically undefined, but in IEEE 754 double precision the tangent of ninety degrees evaluates to about 1.63 times ten to the sixteenth: a large finite number rather than an infinity or a failure. An implementation that does not test for the case therefore returns a confident, wrong, and entirely plausible-looking answer at the pole, the same one for every time of day. No source consulted here discusses this, because it is an artefact of the representation.

What the angles were held to mean

Tradition

Held byPtolemy, Tetrabiblos bk. III ch. 2, second century CE

The rising degree is the anchor from which the rest of the chart’s divisions are laid out, and the accuracy of the whole depends on it before it depends on anything else.

In the Hellenistic material the four angles are the kentra, the pivots, and they carry the strongest emphasis of any position in the chart 2. The place beginning at the Ascendant was the first and most important of the twelve, which is the subject of First House, and the word that names a chart in English descends from the Greek for the rising degree itself.

What none of that settles is whether any of it tracks anything. The angles are the most precisely computable quantities in a chart and the most sensitive to a number, the birth time, that is usually the least precisely known thing about a person. Those two facts sit together uncomfortably: the precision of the computation is real, and it is no evidence about the world.

References

  1. Claudius Ptolemy, trans. F. E. Robbins. Tetrabiblos. Loeb Classical Library 435. Cambridge, MA: Harvard University Press, 1940 (composed c. 150 CE). bk. III ch. 2. find a copy (catalogue search) recordprimary source

    The standard English reference and the chapter-numbering scheme this site adopts and names. Non-Robbins reprints number the chapters differently, so every locus here is given as Robbins numbering.

  2. Chris Brennan. Hellenistic Astrology: The Study of Fate and Fortune: Amor Fati Publications, 2017. ISBN 9780998588902 recordscholarly

    A practitioner-scholar synthesis: comprehensive and well referenced, but not peer-reviewed philology. This site cites it with that characterisation.