Lunar Theory

Law 1: Two Movements of the Moon: The first is "mean" - because it is corrected by the second, giving "true"? Why is the second "mean"? Because of the effects of the sun?

Note: From what and in which direction are the angles measured?

Anomaly (and head?): opposite direction!

TODO MoonHeadMean rambamValue and almagestValue

Rambam uses ‘nimzes’ instead of ‘nimzo’ for 10000 days. Also, Rambam says "'ordered' year"; it was rendered as 254 in accordance with Law 12:1.

Rambam says "'ordered' year"; it was rendered as 254 in accordance with see Law 12:1. Also, this value is actually given in Law 4.

14: 5,6 TODO: Compare with 11:9, about Dli... Something is off!

15:2,3: Corrected Mean Anomaly (Elongation (TODO: ?)). Rambam does not describe the model behind this correction. Because the models Rambam did describe until now are the same as Almagest's models (including numerical parameters), and because this correction is determined by the (doubled) sun-moon elongation, just like Almagest's "improved" model of the moon ("the crank"), I assume that that model was used to calculate this correction. TODO: describe the model!

TODO: Derive the formula! Make the diagram! TODO: How come the fact that sun and moon move in different plains does not affect this calculation?

It seems that Rambam treats "the crank" as just a calculation device - otherwise, table giving visible anomaly from the corrected anomaly should take into account changes in the earth-moon distance depending on the elongation - and it does not! Maybe that is why he does not describe the model behind this correction. And maybe this is why Rambam reiterates - in Law 2 - that only the observability of the new moon needs to be calculated correctly. TODO: How does Almagest calculate visible anomaly?

Law 4,5,6: True Lunar Longitude. Formulae: $$ \tan ^{-1} \left({\sin \alpha \over \cos \alpha + \epsilon} \right) $$ $R$ - radius of the big circle; $r$ - radius of the small circle; $\epsilon = R/r$. A little trigonometry, and we get: visible anomaly = arctg(sin(corrected anomaly)/(cos(corrected anomaly) + e)) and: e = ctg(visible anomaly)*sin(corrected anomaly) - cos(corrected anomaly)

TODO: Make the diagram! Here?

It is clear that values for 150° and 170° are misprinted: they are bigger than the one before them when they should be smaller. Value for 120° is also misprinted, but it is less obvious.

Calculations show that for the precision up to a minute, it is sufficient to know e to up to the second digit after the dot. Below, a column giving the value of e was added to the table. Look at the value of e, and the misprints become obvious: value of e for them is way out of range it is in for the rest of them.

TODO Range of e - graph. Range of possible values for each based on the range of e. This is transcription error: one letter; extra word... Probable correct values.

This is how different sources treat the often misprinted values:

Edition 120° 150° 170°
Vilno incorrect incorrect incorrect
Eshkol incorrect incorrect incorrect
Rambam LaAm corrected in the notes incorrect incorrect
Kapach[a] incorrect corrected in the notes corrected in the notes
Bluming incorrect corrected in the notes corrected in the notes
Frenkel correct and noted correct and noted correct and noted
Keller not given not given not given
Losh[b] incorrect incorrect incorrect

[a] It is not clear how did authoritative Yemenite texts acquire the same misprints as in the ashkenazic editions.

[b] Reprints the text and does not correct any of the misprints - surpising for a textbook on the subject (even introductory). TODO: Link to the scans of the appropriate pages - with the stuff highlighted.