What are “cycles of anomaly” and “cycles of longitude”?

In several early (pre-1600) astronomical texts I read about "cycles of anomaly" and "cycles of longitude", but it us unclear to me what these terms mean. They were clearly familiar to authors at the time and are undefined in any of the texts I've looked through.

How were these terms employed in pre-1600s solar system models? What modern astronomical terms or observable events do they correspond to?

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Perhaps context would help. How are these being used? Today "longitudes" are typical angles as you might imagine, like from the pericenter, to the center or a focus, and then to the object. "Anomalies" are stranger but sometimes more convenient measures, and differ from longitudes when, for instance, orbits are eccentric. They can measure angles to projections of an object to various circular orbits (eccentric and true anomaly) for instance. Or they can be rescalings of swept-out area (= time), as in mean anomaly. – Chris White Sep 24 '12 at 3:10

1 Answer

Cycles of anomaly are lappings of a planet by the Earth. If there are $CA$ of them and the Earth orbited the Sun $N$ times during the time, then one may determine that the planet has orbited $N\pm CA$ times.

Revolutions (not cycles) in longitude are periods between two adjacent nodes, i.e. moments when a planet's trajectory crosses the ecliptic. So it's roughly a revolution of a planet around the Sun but it should be measured by a particular method involving the nodes.

Those concepts are perfectly suited for heliocentric (the modern) system; they are just not being used too often these days.

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I see what confused me, even in a (modern) footnote attempting to explain the term, it says, for example, that "Mars has 37 cycles of anomaly or movements on the epicycle" in 79 solar years. That sounds like it is saying 37 complete revolutions around the epicycle, but it's saying 37 "lapppings" or conjunctions, in which case it makes perfect sense: 37/79 = 2.135, the synodic period of Mars. – raxacoricofallapatorius Sep 25 '12 at 20:27
In the Ptolemaic model each conjunction occurs after $S/P_D$ cycles beyond a full cycle of the epicycle, where $S$ is the synodic period and $P_D$ is the period of the deferent (e.g. the period of the planet, for superior planets). – raxacoricofallapatorius Sep 25 '12 at 20:33
I have another question on Math.SE that I expect you will know the answer to. – raxacoricofallapatorius Sep 25 '12 at 21:40