{"id":1771,"date":"2019-11-25T21:20:20","date_gmt":"2019-11-26T03:20:20","guid":{"rendered":"http:\/\/jonvoisey.net\/blog\/?p=1771"},"modified":"2019-12-01T10:57:15","modified_gmt":"2019-12-01T16:57:15","slug":"almagest-book-iii-table-of-the-suns-anomaly","status":"publish","type":"post","link":"https:\/\/jonvoisey.net\/blog\/2019\/11\/almagest-book-iii-table-of-the-suns-anomaly\/","title":{"rendered":"Almagest Book III: Table of the Sun’s Anomaly"},"content":{"rendered":"

In this chapter, Ptolemy lays out the Table of the Sun’s Anomaly. He actually discusses the table layout at the end of the last chapter, but it seems to me to make more sense to group it with the table so I’ll do that here.<\/p>\n

Ptolemy first notes that the table could be laid out different ways by whe he means you could look up the equation of the anomaly by either the mean motion or the apparent position. However, he prefers to lay out the table by the mean motion.<\/p>\n

The table laid out in 6\u00ba increments within 90\u00ba to either side of apogee, and 3\u00ba increments within 90\u00ba of perigee because the rate of change near perigee is larger than it is near apogee.<\/p>\n

In addition, there’s a symmetry to this table: The equation of anomaly a certain angle of mean motion from apogee is the same as 360\u00ba minus that same angular distance. Thus, Ptolemy only displays this table in 45 lines, but has a second column for angular distance from apogee that is the second half.<\/p>\n

There is one final note of the third column as its sign will depend on which of the first two columns you find the angle in. If it’s the first column, the apparent position will lag the mean, so you should subtract the anomaly from the mean position. If it’s found in the second column, the apparent position will be ahead of the mean position, so it should be added.<\/p>\n

Since this is such a simple table, I’ll post it here in the blog instead of keeping it as a Google sheet.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Angle from Apogee (<\/strong><\/span>\u00ba<\/strong>)<\/strong><\/span><\/td>\nEquation of Anomaly(\u00ba)<\/strong><\/span><\/td>\n<\/tr>\n
6<\/td>\n354<\/td>\n0;14<\/td>\n<\/tr>\n
12<\/td>\n348<\/td>\n0;28<\/td>\n<\/tr>\n
18<\/td>\n342<\/td>\n0;42<\/td>\n<\/tr>\n
24<\/td>\n336<\/td>\n0;56<\/td>\n<\/tr>\n
30<\/td>\n330<\/td>\n1;9<\/td>\n<\/tr>\n
36<\/td>\n324<\/td>\n1;21<\/td>\n<\/tr>\n
42<\/td>\n318<\/td>\n1;32<\/td>\n<\/tr>\n
48<\/td>\n312<\/td>\n1;43<\/td>\n<\/tr>\n
54<\/td>\n306<\/td>\n1;53<\/td>\n<\/tr>\n
60<\/td>\n300<\/td>\n2;1<\/td>\n<\/tr>\n
66<\/td>\n294<\/td>\n2;8<\/td>\n<\/tr>\n
72<\/td>\n288<\/td>\n2;14<\/td>\n<\/tr>\n
78<\/td>\n282<\/td>\n2;18<\/td>\n<\/tr>\n
84<\/td>\n276<\/td>\n2;21<\/td>\n<\/tr>\n
90<\/td>\n270<\/td>\n2;23<\/td>\n<\/tr>\n
93<\/td>\n267<\/td>\n2;23<\/td>\n<\/tr>\n
96<\/td>\n264<\/td>\n2;23<\/td>\n<\/tr>\n
99<\/td>\n261<\/td>\n2;22<\/td>\n<\/tr>\n
102<\/td>\n258<\/td>\n2;21<\/td>\n<\/tr>\n
105<\/td>\n255<\/td>\n2;20<\/td>\n<\/tr>\n
108<\/td>\n252<\/td>\n2;18<\/td>\n<\/tr>\n
111<\/td>\n249<\/td>\n2;16<\/td>\n<\/tr>\n
114<\/td>\n246<\/td>\n2;13<\/td>\n<\/tr>\n
117<\/td>\n243<\/td>\n2;10<\/td>\n<\/tr>\n
120<\/td>\n240<\/td>\n2;6<\/td>\n<\/tr>\n
123<\/td>\n237<\/td>\n2;2<\/td>\n<\/tr>\n
126<\/td>\n234<\/td>\n1;58<\/td>\n<\/tr>\n
129<\/td>\n231<\/td>\n1;54<\/td>\n<\/tr>\n
132<\/td>\n228<\/td>\n1;49<\/td>\n<\/tr>\n
135<\/td>\n225<\/td>\n1;44<\/td>\n<\/tr>\n
138<\/td>\n222<\/td>\n1;39<\/td>\n<\/tr>\n
141<\/td>\n219<\/td>\n1;33<\/td>\n<\/tr>\n
144<\/td>\n216<\/td>\n1;27<\/td>\n<\/tr>\n
147<\/td>\n213<\/td>\n1;21<\/td>\n<\/tr>\n
150<\/td>\n210<\/td>\n1;14<\/td>\n<\/tr>\n
153<\/td>\n207<\/td>\n1;7<\/td>\n<\/tr>\n
156<\/td>\n204<\/td>\n1;0<\/td>\n<\/tr>\n
159<\/td>\n201<\/td>\n0;53<\/td>\n<\/tr>\n
162<\/td>\n198<\/td>\n0;46<\/td>\n<\/tr>\n
165<\/td>\n195<\/td>\n0;39<\/td>\n<\/tr>\n
168<\/td>\n192<\/td>\n0;32<\/td>\n<\/tr>\n
171<\/td>\n189<\/td>\n0;24<\/td>\n<\/tr>\n
174<\/td>\n186<\/td>\n0;16<\/td>\n<\/tr>\n
177<\/td>\n183<\/td>\n0;8<\/td>\n<\/tr>\n
180<\/td>\n180<\/td>\n0;0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

While that’s the end of the chapter, I do want to quickly look at this table in another form. Specifically, let’s look at its graph:<\/p>\n

\"\"<\/a><\/p>\n

Here, I’ve done as Ptolemy has stated and taken the values from the first column as subtracted, and the second column as additive. One of the things we can take away from this is the points where the apparent speed equals the mean. Speaking in modern calculus terms for a moment, this is where the first derivative = 0. For non-calculus types, that’s just the points where things turn around and change from going down to going up or vice-versa. This happens to be at 90\u00ba and 270\u00ba. Tuck that away in your mind as we’ll be using it in a few posts.<\/p>\n

\n

\"\"<\/a><\/p>\n

\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

In this chapter, Ptolemy lays out the Table of the Sun’s Anomaly. He actually discusses the table layout at the end of the last chapter, but it seems to me to make more sense to group it with the table so I’ll do that here.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[24],"tags":[25,10,14],"class_list":["post-1771","post","type-post","status-publish","format-standard","hentry","category-almagest","tag-almagest","tag-geocentrism","tag-ptolemy"],"acf":[],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p9ZpvC-sz","_links":{"self":[{"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts\/1771","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/comments?post=1771"}],"version-history":[{"count":4,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts\/1771\/revisions"}],"predecessor-version":[{"id":1801,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts\/1771\/revisions\/1801"}],"wp:attachment":[{"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/media?parent=1771"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/categories?post=1771"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/tags?post=1771"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}