Almagest Book IV: Table of Lunar Equation of Anomaly (First Anomaly)

Although I don’t see any particular reason we couldn’t have done this several chapters earlier, Ptolemy has elected to hold off producing his table of Lunar Anomaly until almost the very end of Book IV1.

At the end of the previous chapter, he did toss in a few expository notes on this table noting that this will be useful in the future for “calculations concerning conjunctions and oppositions”2. To do so, Ptolemy doesn’t show his math, but notes that it was done “geometrically, in the same way as we already did for the sun.”

However, to use that method requires knowing the ratio of the radius of the deferent to that of the epicycle. Ptolemy states he takes a ratio of $60:5 \frac{1}{4}$. Using that, he calculates his table in the same manner as he did for the sun, which is in $6º$ intervals near apogee and $3º$ intervals near perigee. However, because this epicycle rotates opposite the direction of the deferent, we should subtract if the angle is in the first column, and add if it is in the second3

That being laid out, the table is below:

Angle from Apogee (º) Equation of Anomaly(º)
6 354 0;29
12 348 0;57
18 342 1;25
24 336 1;53
30 330 2;19
36 324 2;44
42 318 3;8
48 312 3;31
54 306 3;51
60 300 4;8
66 294 4;24
72 288 4;38
78 282 4;49
84 276 4;56
90 270 4;59
93 267 5;0
96 264 5;1
99 261 5;0
102 258 4;59
105 255 4;57
108 252 4;53
111 249 4;49
114 246 4;44
117 243 4;38
120 240 4;31
123 237 4;24
126 234 4;16
129 231 4;7
132 228 3;57
135 225 3;46
138 222 3;35
141 219 3;23
144 216 3;10
147 213 2;57
150 210 2;43
153 207 2;28
156 204 2;13
159 201 1;57
162 198 1;41
165 195 1;25
168 192 1;9
171 189 0;52
174 186 0;35
177 183 0;18
180 180 0;0


 

  1. Seriously. Only one more chapter to go after this!
  2. Sounds a lot like predicting eclipses to me.
  3. Noting that the angle is measured in the direction of rotation which is to say, clockwise.