{"id":2133,"date":"2020-08-09T18:21:36","date_gmt":"2020-08-09T23:21:36","guid":{"rendered":"http:\/\/jonvoisey.net\/blog\/?p=2133"},"modified":"2020-08-09T19:13:50","modified_gmt":"2020-08-10T00:13:50","slug":"almagest-book-iv-alexandrian-eclipse-triple-solar-lunar-positions-epicyclic-anomaly","status":"publish","type":"post","link":"https:\/\/jonvoisey.net\/blog\/2020\/08\/almagest-book-iv-alexandrian-eclipse-triple-solar-lunar-positions-epicyclic-anomaly\/","title":{"rendered":"Almagest Book IV: Alexandrian Eclipse Triple – Solar\/Lunar Positions & Epicyclic Anomaly"},"content":{"rendered":"

Modern commentary on Ptolemy often downplays the\u00a0Almagest<\/em> because it is certainly a work that relied heavily on the work that astronomers before him. While we no longer have a thorough record of those predecessors, it seems that few historians think much of the\u00a0Almagest<\/em> was truly novel<\/span>1<\/sup><\/a><\/span>. But I would hasten to remind that, while Ptolemy stood on the shoulders of those who came before, he certainly climbed there on his own, not simply accepting their results, but doing his best to validate them.<\/p>\n

And we’re about to get a big dosing of that, because all the work we’ve done in the past three posts, we’ll be redoing with a new set of eclipses observed by Ptolemy himself, allowing for an independent check on the important value of the radius of the epicycle.<\/p>\n

To get us started, let’s look at the eclipses Ptolemy observed:<\/p>\n

The first occurred in the seventeenth year of Hadrian, Pauni [X] 20\/21 in the Egyptian calendar [133 May 6\/7]. We computed the exact time of mid-eclipse as $\\frac{3}{4}$ of an equinoctial hour before midnight. It was total. At that time, the true position of the sun was about $13 \\frac{1}{4}\u00ba$ into Taurus [$43 \\frac{1}{4}\u00ba$ ecliptic longitude].<\/p>\n

The second occurred in the nineteenth year of Hadrian, Choiak [IV] 2\/3 in the Egyptian calendar [134 Oct 20\/21]. We computed that mid-eclipse occurred 1 equinoctial hour before midnight. [The moon] was eclipsed $\\frac{5}{6}$ of its diameter from the north. At that time, the true position of the sun was about $25 \\frac{1}{6}\u00ba$ into Libra [$205 \\frac{1}{6}\u00ba$ ecliptic longitude].<\/p>\n

The third eclipse occurred in the twentieth year of Hadrian, Pharmouthi [VIII] 19\/20 in the Egyptian calendar [136 Mar 5\/6]. We computed that mid-eclipse occurred 4 equinoctial hours after midnight. [The moon] was eclipsed half of its diameter from the north. At that time the position of the sun was about $14 \\frac{1}{12}\u00ba$ into Pisces [$344 \\frac{1}{12}\u00ba$ ecliptic longitude].<\/p><\/blockquote>\n

Now we’ll determine the solar motion and the period for these pairings. From the first to the second, this is a change of 161;55\u00ba and takes place over the course of 1 year, 166 days, and $23 \\frac{5}{8}$ equinoctial hours if converted to solar days.<\/p>\n

Doing the same for the second eclipse to the third we get a change of 138;55\u00ba in solar position in a period of 1 year, 137 days, and $5 \\frac{1}{2}$ equinoctial hours.<\/p>\n

Next, we turn to the lunar mean motion table<\/a> and look up the change in longitude corresponding to the intervals in question in both longitude (motion on the deferent) and anomaly (motion around the epicycle). Doing so we find that for the first to the second eclipse we have an increase of 169;37\u00ba in ecliptic longitude and 110;21\u00ba in anomaly. For the second to third eclipse, we find an increase of 137;34\u00ba in ecliptic longitude and 81;36\u00ba in anomaly.<\/p>\n

This then gets compared to the solar motion<\/span>2<\/sup><\/a><\/span>. Thus, for the first to second eclipse, the 110;21\u00ba motion around the epicycle produces a 7;42\u00ba difference<\/span>3<\/sup><\/a><\/span> in ecliptic longitude. For the second to third eclipse, the 81;36\u00ba motion around the epicycle produces a 1;21\u00ba difference<\/span>4<\/sup><\/a><\/span> in ecliptic longitude.<\/p>\n

Up next, we’ll again walk through the process of determining the radius of the epicycle to make sure it lines up with the values from Hipparchus!<\/p>\n

\n

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

\n

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

Modern commentary on Ptolemy often downplays the\u00a0Almagest because it is certainly a work that relied heavily on the work that astronomers before him. While we no longer have a thorough record of those predecessors, it seems that few historians think much of the\u00a0Almagest was truly novel. But I would hasten to remind that, while Ptolemy … <\/p>\n

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Solar\/Lunar Positions & Epicyclic Anomaly”<\/span><\/a><\/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":[25,46,19],"class_list":["post-2133","post","type-post","status-publish","format-standard","hentry","category-almagest","tag-almagest","tag-eclipse","tag-moon"],"acf":[],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p9ZpvC-yp","_links":{"self":[{"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts\/2133","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=2133"}],"version-history":[{"count":4,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts\/2133\/revisions"}],"predecessor-version":[{"id":2142,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/posts\/2133\/revisions\/2142"}],"wp:attachment":[{"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/media?parent=2133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/categories?post=2133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jonvoisey.net\/blog\/wp-json\/wp\/v2\/tags?post=2133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}