The following solutions have been on the record for years. Many have appeared in highly prominent journals and forums, including DIO. But most of the history-of-astronomy political center (i.e., the gang that controls funding) is not technically and-or emotionally capable of coping with the obviousness of the triumphs of long-resented unorthodoxy, achieved by those whose scientific skills have so often embarrassed that selfsame immovable centrist cult. (DIO's board, judges and prizewinners boast all of the handful of highly skilled mathematicians, physicists, and statisticians presently doing topflight scientific history of ancient astronomy.)
DIO's failure to enlighten the invincibly-unenlightenable will not be surprising to anyone who has tracked (or tried to track) the tragicomic avalanche of foulups by snobster-centrist forums (e.g., the pseudo-refereed JHA), revealed year after year in DIO & the Journal for Hysterical Astronomy. (See, e.g., the funnies revealed most recently in DIO 16  ‡1 & ‡3 and below. Fuller catalog on separate posting.
But to see the dim determined to STAY (or act) heart-wrenchingly dim is a needless spectacle, since recent competing displays — e.g., the White House's Dork Ages 2001-2009 — have rendered such demonstrations superfluous.
Even worse than dimness: instances of bizarre and equally cemental (ibid §E4 [p.26]) centrist misjudgements, even by the political center's very best minds. (Scholars bright & valuably expert in certain areas — though just as cementally incapable of publicly admitting same of the DIO-tribe.) See, e.g., Alex Jones (in a paper whose incomprehension is incomprehensible [scholarship-wise]: ibid §F3 [p.28], also DIO 5  n.26 [p.9]); as well as John Britton's clinging undislodgeably to his longtime project of fantasy-speculating pointlessly massive, laborious, cumbersome, and inevitably imprecise Babylonian method (to explain ancient lunar speeds), supported by not-a-shred of attestation or numerical path-to-lunar-speed reconstructions. (By contrast, DIO's three solutions of Britton's mysteries apply methods which are simple & solidly attested in antiquity, thereby deftly and precisely recovering all 24 sought digits [of the lunar speeds in question]. On the nose in every case. Summation at DIO 16  p.2.)
Several examples of fruitful DIO inductions
are provided below.
(See also DIO's $1000 challenges at DIO 11.2  p.33. Also the long list of eminent-historian slips & bungles at DIO 4.1  ‡4 §A [p.48], none retracted by anyone but NYU's Alex Jones — an imaginative & sometimes productive speculator who, however, seems unable to grant us more than one concession of fallibility.)
[It should also be noted that one utterly undeniable DIO discovery is assented to even by some cultists (though never by the JHA's seething top politicians): the revolutionary finding that the Babylonian tropical yearlength on Babylonian text BM55555 was based upon two well-known Greek observations. The discovery has been honored by this cuneiform text's permanent public display at the British Museum.]
The counter-progressive aspect of broad cult-stubbornness here is rendered most blatantly and pathetically obvious by the unappreciated fruitfulness of the proscribed findings, which is the theme to be especially emphasized in what follows.
Examples of DIO Inductive
The always-ridiculous 1976-2002 controversy
over the authorship of the Ancient Star Catalog
(Almajest 7.5-8.1) has been over for nearly a decade.
(The centrist Journal for the History of Astronomy no longer wastes dozens of Pb pages fighting the obvious. But it has never admitted its mistakes in both science and slander.)
Realization of the truth led to explanations of several formerly mysterious oddities of the Catalog, e.g.:
Glaring excesses of 40' and (especially!) 10' endings in Catalog longitudes. (R.Newton Crime of Claudius Ptolemy Johns Hopkins Univ 1977.)
Absence of a half-degree amplitude wave that's rigorously implied by Ptolemy's formerly-presumed observership. (DIO 2.3  ‡8 §C22 [p.110].)
The lack of appearance of Ptolemy's −14' error in his own latitude. (Ibid.)
Several other striking anomalies. (See the discoveries of K.Pickering: DIO 12  ‡5 [pp.59-66].)
Aubrey Diller-DR solution for the Hipparchos-Strabo klimata table.
(DIO 4.2  p.56 Table 1; DIO 5  Table 0; DIO 16  Tables 1&2, Fig.1.) The theory's productivity is obvious from DIO 16  ‡3 eq.3 [p.21] or DIO 5  n.25 [p.9]. Its predictive confirmations (since Diller's discovery) are numerous and international: ibid §D3 [pp.8-9].
DIO 11.3's 2002 Duke-DR-Thurston overturning of Gingerich's 2002 Venus-made-me-do-it alibi for Ptolemy's forgery of the “observations” he pretended to found his Venus theory upon. One subsequently-discovered tip-off of the forgery is the quarter-degree endings of Almajest stars associated with Venus. See DIO 11.3  ‡6 §§C1-2 [p.76].
Eratosthenes & Poseidonios Earth-Sizes: hugely disparate
(former 40% larger than the latter),
yet both's errors (+19%, −17%, respectively) are neatly explained
— both to within 1% —
by the same precisely fruitful theory:
distortion due to horizontal light-rays' atmospheric-refraction-curvature,
which equals 1/6 of Earth's curvature.
DIO 14  ‡1 eq.28 [p.11].) This led on to realization that the famous Pharos lighthouse at Alexandria was the source of Eratosthenes' famous Earth-size: DIO 14  ‡1 [pp.3-12].
Both hitherto-mysterious Hipparchos lunar numbers, 3144 & 3122 1/2,
are found to have a startlingly cohesive
— and historically revealing — common solution,
to the full (extremely high) precision displayed.
(DIO 1.3  §§O2&O3 [pp.160-161] eqs.23&24. See also H.Thurston's intelligent discussion of this finding in the History of Science Society's Isis — vol.93 #1 pp.58-69 .)
Ptolemy's equation 8523 Metonic years = 105416 months has been shown (DIO 6  ‡1 §§I10-14 [pp.23-24]) to derive from the valid (and anciently observed) eclipse cycle 781 sidereal years = 9660 months, using a Hipparchan precession equation (attested in the same Ptolemy paragraph containing the 8523 year relation). This finding's fruits included: [a] Verification of ancient use of eclipses for accurate celestial speeds; [b] Realization that the 160y cycle of Geminos 8.40-41 was simply 1/5 of the 800y eclipse cycle which centered a nest of long sidereal-year eclipse cycles (743y', 762y', 781y', 800y', 819y', 838y'). [c] Ancient use of the hitherto-merely-theorized transformation between sidereal & relations via the Ptolemy-attested relation 35999 sidereal years = 36000 Metonic years.
The errors of Erathosthenes' famous S.Solst ZD, obliquity, Alexandria latitude are all multiples of 8'; so the single theory that the 1st was measured by gnomon, all three errors vanish. [First published in Isis 1982.]
DR showed that Pliny's circuli came from a clever ancient linear fit to Mediterranean data computed from sph trig. This theory then led on to a perfect solution of Pliny's otherwise wildly disparate Rhodos klima. And see DIO 16  ‡3 §I1 [p.32] & n.50 [p.35].
Solution of Hipparchos' mis-latitude of Carthage is consistent with same odd explanation for Alexandria's mis-latitude — in the very same paragraph of Strabo. (The fruitful theory in this case is originally due to E.Honigmann, and was commendably backed by O.Neugebauer HAMA 1975 p.336 n.29, though ON's cult has now found it necessary to reject it (DIO 16  ‡3 §F4 [pp.28-29]) in order to repulse acceptance of [a] DIO's Carthaginian expansion of it, and [b] Diller-DR's priority in proving Hipparchos' possession of sph trig in the 2nd century BC.
Aristarchos' exeligmos (57y + 32°) is found consistent with both his Great Year solution and his possession of the precise Babylonian month. (Alter Orient und Altes Testament 297:295; and simultaneously DIO 11.1  ‡1 eqs.12&13.)
DIO 1.3  (n.288 [p.173]) proposed that Hipparchos' grossly disparate lunar-eclipse placements of Spica (Almajest 3.1) had been corrupted by his having mis-signed the parallax corrections. This theory brings the two data (previously disagreeing by over 1°) into nearly perfect accord. It was later found that Hipparchos' star catalog's worst fundamental star longitude, Regulus, is explainable likewise. DIO 16  ‡3 [pp.3-12].) All three data's errors thereby descend from ordmag 1° to ordmag 1'. Error repeated modernly (see idem) by JHA now-Assoc-Ed James Evans.