Finding that Eratosthenes' errors in his obliquity and his Alexandria-latitude were each 1/2 of the solar semi-diameter, DR hypothesized that Eratosthenes had found both together by the standard method of Almajest 1.12, using his semi-diameter-corrupted gnomon observation of the S.Solst ZD (360°/50) with a W.Solst observation taken by a competent astronomer on a transit circle. Eratosthenes' errors in his S.Solst ZD, latitude L, and obliquity ε are, resp: −16', −8', and +8'; but, on this theory, all drop to virtually null. (DR Isis 73:259-265 [1982]; DIO 16 [2009] ‡3 n.30 [p.29].)
When DR showed in 1985 that the yearlength on Babylonian Cuneiform Text 210 (British Museum BM55555) was based on Greek solstices (Meton & Hipparchos), this provided at last (DIO 1.1 [1991] ‡6 eq.6 [p.51]) the hour (dawn) of Hipparchos' −134/6/26 1/4 S.Solstice.
Which gave also (ibid eq.8 [p.53]) the hour of Aristarchos' −279/6/26 1/2 S.Solst. These being the only two cardinal-pt solar events that Almajest 3.1 cites without giving the hour (ibid §B5 [p.52]), it became obvious (see, e.g., A.Jones Wrong for All the Right Reasons 2005 pp.23f) that Ptolemy's reticence was due to each's absolute value being 1d/4 offset from the time given by Ptolemy's usual solar theory, Hipparchos' prime “PH” orbit. (Each's offset was in the same direction, thus providing differential consistency with PH. See DIO 20 [2012] ‡3 §P5 [p.18].) Aristarchos' S.Solst time was merely the usual day-epoch-truncated calendaric value, but Hipparchos' SS agreed with reality within 1h, providing the 1st empirical solstice yet reconstructed from antiquity. (This and the only three others yet knowable are found at DIO 20 [2012] Table 3 [p.17].)
The −134 Hipparchos S.Solst established via BM55555, merged with the equinox data given by Almajest 3.1, led on to reconstruction of Hipparchos' ultimate solar orbit, “UH”, from around −134. The UH orbit evaporated the mystery of the 1°/4 discrepancies between the PH orbit (long Hipparchos' standard, later used by Ptolemy) and 2 of the 3 calculated solar longitudes of Almajest 5.3&5: all 3 longitudes had been calculated instead from the UH orbit, the agreement being to 1' in all 3 cases.
One of the three UH-computed longitudes of Almajest 5.3&5 nearly agreed with PH only by fortunate accident, so that Ptolemy exceptionally did not recalculate it via PH, with the invaluable result that Hipparchos' calculation of its mean longitude was not tampered-with. It disagrees with PH by nearly 1°/10 but agrees with UH precisely (DIO 1.1 [1991] ‡6 §§H5-H6 [p.64]), a perfect verification of theory and (as Alex Jones points out to less intelligent colleagues) 2nd century BC use of mean longitude.
From rounding considerations, it was then realized (DIO 1.1 [1991] ‡6 eq.28 [p.58]) that the UH orbit's mean-longitude-at-epoch was −126/9/24 1/2, which is 16 Metonic cycles after Meton's SS.
The interval is 304y1/4. It was later realized (DIO 11.1 [2002]) ‡1 n.17 [p.9]) that this is exactly 1/16 of Aristarchos' Great Year of 4864y.
Such turned out to be the 1st hint of what was finally revealed
as an astonishing, hitherto-unknown geometric series of
embeddings of cycles in the 4868y Great Year.
304y1/4: 1d difference between Kallippic
& Hipparchos “tropical” calendar;
608y1/2: saros-cycle return to same longitude;
1217y: and with solar return;
2434y: and with lunar return;
4868y: and with diurnal return.
(DIO 11.1 [2002]
nn.14&17 [pp.8-9].)
It had long been a mystery that while the amplitude of the periodic error in the PH solar theory was 0°.4, the periodic-error amplitude in the zodiacal longitudes of the Ancient Star Catalog was 0°.2. The mystery vanished in light of DR's discovery of the UH orbit, whose periodic-error amplitude was 0°.2. (DIO 1.1 [1991] ‡6 §F3 [p.61].)
DIO 7.1 [1997]
contended in 1997 (contra two experts who'd served time
at the Princetitute) that since Aristyllos' six extant star declinations
(Almajest 7.3) were all consistent with
quarter-degree rounding, then Hellenistic astronomers were expressing
angles in sexagesimal measure as early as the 3rd century BC,
even though when publishing they might express their results
in the traditional circle-fraction format so dear to the era's pedants.
(DR Isis 1982.) Years later (2008) DR applied this suspicion to
Archimedes' odd-looking solar-size brackets (Sandreckoner):
Sun's diameter between rt.angle/200 and rt.angle/164. This revealed
(DIO 20 [2012]
‡1) that the original measurement was:
solar diameter between 27' and 33'.
[For the last 2300y, anyone who got through the 8th grade could have
solved this mystery. As far as we know none did until DIO.]
In 1982 (Isis) DR derived a non-traditional Eratosthenes Earth-circumference 256000 stades from E's Nile Map. In 2008, DR discovered that Eusebius had preserved E's adopted Earth radius: 40800 stades. Multiplication by 2π yields 256000 stades. (See DIO 14 [2008] ‡1 eq.18 [p.8].)
The math doesn't work in reverse (ibid eq.17), so determination of radius preceded that of circumference, which [a] tells us that the famous “Eratosthenes Experiment” is a fable, and [b] favors the Lighthouse Method as the actual source of Earth-radius R.
Accounting for the 1st time for the fact that tall ships' masts were very roughly 1/4 as high as the Lighthouse, DIO 14 [2008] ‡1 §C [pp.5-6] reasoned from Josephus' testimony (that the flame was visible from up to c.300 stades distance), DR computed that Sostratos' Alexandria Lighthouse flame was approximately 300 feet high — and then figured it was reasonable that such a spectacular record would be proudly precise, so the Pharos might be exactly 300 feet. This implied (ibid eqs.2&21 [pp.5&9]) a remarkable result, an accident of stade-measure: the Earth's radius could be found by simply squaring the number of stades the flame could be seen over the water. Squaring 202 and rounding to the traditional stade-hundreds yields Earth-radius R = 40800 stades, which is just the Eratosthenes value established by Eusebius and by the Nile Map.
And so our investigation of R's source has handed us a wonderful spinoff-extra: solution of the longstanding mystery of how high the Lighthouse flame was: 300 feet precisely and deliberately.
A further now-obvious but novel result issuing from the foregoing: the Earth-size historically attributed to Eratosthenes was actually due to Pharos-architect Sostratus.
Very shortly after posting the foregoing for refereeing, DR found that a 12th century Arabic report made the Pharos flame 50 fathoms high, which is 300 feet. (Ibid p.12.)
The 40800 stades value for R is 19% HIGH, which is close to the plus-20% error one would expect from the lighthouse method of measuring the Earth, since the Earth's atmosphere bends horizontal light with 1/6 of the curvature of the Earth. The other stay-at-home method of measuring the Earth is the Double-Sunset Method (DR Amer.J.Physics 1979) which is also affected by a factor of 6/5 due to atm refraction, but divisively not multiplicatively, thus predicting a value 17% LOW.
The only other value known from antiquity (Ptolemy used both successively) is Poseidonios' circumference, 180000 stades — which is 17% low. Thus, the atm refraction theory explains both standard ancient Earth-size values to within 1%. An ideal-textbook instance of fruitfulness.
DIO-Journal for Hysterical Astronomy 1.1 [1991] ‡7 §C1 [p.69] 1st published the proposal — based upon the discrimination-limit of the human eye being about 1/10000 of a radian — that Aristarchos' famous 87° figure for half-Moon lunar elongation was a lower limit, not a precise value. Thus, this canonical datum was finally quantitatively explained. (Details: DIO 14 [2008] ‡2 n.17 [p.17].)
If 87° was a lower-bound, then Aristarchos contemplated that the ratio of the Sun's distance to the Moon's was anywhere in the range from 19 to ∞ — a point that was later found to provide an explanation of why pseudo-Aristarchos' lunar distance was 19 Earth-radii, not 20.
The 1/10000 radian limit to human vision, having led DR to the solution for Aristarchos' half-Moon elongation, was next applied to his scale of the universe — and it produced the very solar distance which we find in Archimedes' analysis of Aristarchos' researches on the subject: DIO 14 [2008] ‡2 §F1 [p.26].
Likewise, when the same theory was tested on Aristarchos' estimate of the stars' distance, it predicted 10000 AU, and this is testified to by Archimedes: DIO 14 [2008] ‡2 §§B1&E2 [pp.17&25].
Thus, all three of Aristarchos' data towards finding the universe's scale turn out to be based upon the same 1/10000 radian estimate of human vision: DIO 14 [2008] ‡2 §F9 [p.28]. Which backs up — and explains the significance of — ancient testimony that Aristarchos studied human vision: ibid §B2 [p.18].
All of which supports — as does our demonstration of the precision of Sostratos' measure of the Earth's size — our journal's inaugural proposal (DIO 1.1 [1991] ‡1 n.24 [p.10]) that the common assertion (ibid §A [pp.13-17]) that Greek students of the heavens were non-empirical was a false mass-slander.
When DR showed (New York Times 1996/5/9 p.1) that Byrd had faked the last leg of his alleged flight to the North Pole on 1926/5/9, the prime evidences were contradictions in Pole-arrival-times & en-route solar sextant altitudes. The judgement was confirmed when DR discovered the killer-evidence (which NGS and DR had previously missed) that Byrd's sextant data precision was in arc-seconds, a precision utterly impossible on his sextant. (DIO 10 [2000] §G6 [pp.39-41].) Obviously, whoever had indoor-manufactured the trip's “observations” had forgotten to round them to the sextant's actual half-arcmin precision (found in the many real outdoor observations prior to 1926/5/9 in Byrd's diary record).
Further confirmatory: between his 1926/6/22 transmission of his over-precise numbers (to the Navy & NGS) and Autumn, Byrd discovered his blunder and deleted all the suspicious raw data before sending the remainder — merely processed (“reduced”) solar altitudes — to the American Geographical Society on 1926/11/24. (Sample textual before-deletion vs after-deletion comparison at DIO 10 [2000] Fig.7 [p.34].)
When an explorer starts faking a portion of a trip, he gets cautious
about committing contemporaneous facts to paper.
Henshaw Ward and DR noted the significance of the fact that
when R.Peary reached his final “North Pole” camp,
he stopped writing in his diary for days. When fellow N.Pole hoaxer
R.Byrd's 1926 diary was de-sequestered in 1996,
DR found that the same caution turned up:
Byrd kept a chatty diary virtually daily since leaving NYC 1926/4/5;
but following his 5/9 falling short of the N.Pole, he stopped writing in it
for the rest of the year.
(DIO 10 [2000]
p.105.)
[F.Cook's behavior follows the same pattern,
as shown by the ongoing researches of R.Bryce.]
DIO 1.3 [1991] proposed a Hipparchos −157 “EH” solar orbit which drew from his PH-discrepant observed −157 Autumnal Equinox, a regular −157 Alexandria-krikos Vernal Equinox observation, and a −157 Summer Solstice not observed but computed indoors from Kallippos' calendar (since Hipparchos evidently did not yet know how to observe a solstice: DIO 20 [2012] ‡2 §L [p.14]). Such an orbit turned out to completely explain the hitherto-mysterious solar longitudes of the 2nd (Trio B) of the two Hipparchos eclipse trios reported at Almajest 4.11. (DIO 1.3 [1991] §L2 [p.14].)
Two decades after 1991, Anne Tihon discovered a previously unknown Hipparchos S.Solst: −157/6/26 21h AAT. (First drawn to DR's attention by Dennis Duke.) The just-cited 1991 theory, that he had (during the same year, −157) indoor-computed the EH S.Solst, suggested the obvious parallel theory that he may have tested using the calendar of Meton instead of Kallippos. When this idea was checked, the calculation's agreed with the Tihon (hour-precision) S.Solst within 6 time-minutes! (DIO 20 [2012] ‡2 eq.25 [p.15].)
The calculated solar longitudes Hipparchos' 1st eclipse trio (Trio A) turned out to be based on a merge of the outgoing EH (−157) and incoming PH (−145) orbits — and in precisely the shares expected during the transition from EH to PH.
Which dates his Trio A calculation to about −145 when the PH orbit was established. (DIO 20 [2012] §O [pp.16-17].)
The new PH elements that were simple constants (apogee & mean-longitude-at-epoch) had been adopted immediately, but adoption of those PH elements (mean motion & eccentricity) that required tables (which would take at least months of computation to prepare) had to be delayed. So, already-existing tables based on the old EH elements (mean motion & eccentricity) were used temporarily (until the two needed PH tables were ready). DR called this stitched-together set of elements the Hipparchos “Frankensteinorbit”. (DIO 1.3 [1991] §M [pp.146-149].)
In DIO 1.3 [1991] DR showed that Hipparchos had analysed his eclipse trios in pairs not — as had been universally believed from Ptolemy (Almajest 4.5) to the present (Toomer, Neugebauer, etc) — threesomes. (DIO 20 [2012] ‡3 eqs.9-11 [p.24].)
And R.Newton 1977 had realized that Trio A contained a discrepancy of 1° in the 3rd eclipse. In 2012, DR found the cause of the 1° fudge (DIO 20 [2012] ‡3 §G1 [p.25]): it had enormously reduced the discrepancies in Hipparchos' three pair-solutions for Trio A. But the result was a faked 179° true-longitude lunar-elongation mid-eclipse, a physical impossibility (ibid §G4 [p.26]).
The Pair Method (ibid eqs.16-18 [p.155]) solved Trio A's lunar e: ibid eq.19 [p.156]. The very same method then solved Trio B's r: ibid eq.20 [p.156]. Both solutions used the same lunar mean-longitude-at-epoch, Hipparchos-Ptolemy's 178° (eqs.8&9 [pp.151&153]); and the same mean-anomaly-at-epoch, Hipparchos' 82° (§N12 [p.154]) — the tabular epoch being the traditional Phil 1 (sometimes called “the death of Alexander”, e.g., Almajest 3.1). Each of these two elements was an INTEGRAL number of degrees: classic ancient rounding practice.
During a talk at the British Museum in 2001, DR broached the theory (DIO 11.1 [2002] ‡1) that Aristarchos (about 280BC) originated the “Babylonian” month, using Ptolemy's equation (Almajest 4.2)
but treating it via the Kallippic year to find (to amazing precision)
where S = saros, M = month, K = Kallippic 365d year.
Which produces
(thanks to a small rounding pointed out to DR by John Britton and John Steele:
DIO 11.1 [2002]
‡1 §A8 [p.7])
the exact “Babylonian” month.
The next fruit of the eclipse-cycle hypothesis was swift solution of the long-mysterious source of the System A month: it is simply half of a long eclipse cycle, using eclipse data from the 13th century BC. See DIO 11.1 [2002] ‡2.
The last of the Greek-era eclipses usable to find this relation was −262/1/26 (ibid §B4 [p.13]). The 1st sure System A cuneiform tablet computes full Moons from late −262 to late −251 (ibid §E6 [p.18]; Neugebauer ACT 1955 1:117, 3:47).
Yet further fruitfulness for the theory that the ancients' highly accurate lunar speed values were based on huge eclipse cycles: DIO 11.1 [2002] ‡3 found that Hipparchos in −140 discovered the accurate draconitic equation
by using eclipses from −1244/11/13 (near apogee) and −140/1/27 (near perigee), 1103y apart. The equation peculiarly required (see idem) a perigee-apogee pairing; and the only astronomer ever to use such for a draconitic equation was Hipparchos — indeed, he is attested (Almajest 6.9) as doing so for the same −140/1/27 perigee eclipse, for the very same purpose (over a shorter [but still large] interval). The cited 1103y cycle's recovery of the eight digits (of the above equation) is perfect for all eight.
Fortunately, A.Jones immediately objected that six Babylonian cuneiform texts were using this equation already from c.200BC (long before Hipparchos). This led to the DR discovery that this Neugebauer allegation is not supported by the actual fine details of his own 1955 book (ACT): of the 6 such ACTs, half are undated (could have been computed any time) and these are the only lunar latitude ACTs calculated for c.200BC that use the equation in question; the other three ACTs, dated-on-the-clay to c.200BC, don't use it. There is a 7th ACT that is dated and does use it, but it is from decades after Hipparchos. The upshot is 7-for-7 consistency with the theory that the equation is Hipparchos', as Ptolemy said (Almajest 4.2).
The unexpected fruitfulness here (of a theory originally aiming at solving lunar speeds, not cuneiform-tablet-dating) is production of a precise 7-out-of-7 warning (effectively if reluctantly co-authored by Jones) that undated ACTs are not to be assumed to have been calculated (any more than modern astronomical calculations) at the date for which the work is done. See DIO 11.1 [2002] ‡3 §D2 [p.23].
In 2003 DR found (DIO 13.1 [2003] ‡2) yet another vindication of the eclipse-cycle source of lunar speeds, when he found it possible to explain Ptolemy's final equation 3277u = 3512v (PlanHyp 1.1.6) by using a 1325y eclipse-cycle (the longest of all those used by the ancients):
(where u = synodic months, v = anomalistic month, w = draconitic month, g = anomalistic year).
But fruitfulness was not only for the eclipse-cycle theory. The astonishing aspect of the foregoing three cycle-solutions is that: all three earlier eclipses were from merely a one-century range — despite the fact that the later (classical Hellenistic era) eclipses paired with them were from a range sprawling across nearly half a millennium: from the early 3rd century BC to the late 2nd century AD. This strongly suggests that the earliest data available to Hipparchos and others was from a specific tight time-period: the 13th century BC.
It is gratifying to find that this is known to be
around the time astronomical observations began in Babylonia
(Isis 83:474 (1992): c.1350BC. See
DIO 16 [2009]
p.2 for summation of this and the following 3 facts:
All three anciently-adopted lunar-speed values newly solved by us
(DIO 11.1 [2002],
DIO 13.1 [2003])
are accurate to 1 part in ordmag a million or better.
(As are the System B synodic & anomalistic months already known
from Almajest 4.2 to derive from an eclipse cycle.)
All three of these speeds are expressed in the ancient sources as ratios of four-digit integers: 24 digits, which is what is to be expected for results obtained by eclipse ratios.
All of these attested 24 digits are EXACTLY matched by DIO, merely through division of an eclipse cycle by a tiny integer (or half of such).
DIO 4.2 [1994] p.58 published an improvement of Aubrey Diller's 1934 sph trig solution of the Hipparchos-Strabo klimata, showing the startling enhancement of Diller's fit if latitudes had been rounded conventionally to the nearest 5' (1°/12). This theory solved the 1st of Diller's seeming failures. Thus encouraged, DR tried it on other klimata. Ultimately, all fit. This improved version of Diller's discovery was henceforth known in DIO as the Diller-DR Theory.
Some established klimata not known to Diller were tested (idem): all agreed on the nose.
The one non-fit had always been Meroë. But on 2009/4/1, DR found that it fit after all, and posted the new find on 4/6 in DIO 5 [2009] n.25 [p.9]. Math at DIO 20 [2012] ‡3 eq.3 [p.21].
A new klima, Cinnamon Country (12h3/4, 8800 stades) was investigated in DIO 5 [2009]. The Diller-DR Theory fit it precisely.
At DIO 16 [2009]
‡3 Fig.1 [p.24],
the Diller-DR solution is graphed, dramatically
exhibiting its extreme sensitivity to the slightest variation
in the sph trig solution Diller left us. Two more klimata
(Meroë, Cinnamon, Equator) brought the total number of klimata
to fourteen, and the new ones fit just as perfectly as the previous eleven.
The proposed theories of Neugebauer and of Jones both fail at
(among other klimata) the Equator, while Diller-DR fits it perfectly.
Which makes that theory's score: 14-for-14. Not counting the Equator,
DIO 16 [2009]'s
cover asked:
DIO 1.3 [1991] n.288 [p.173] examined J.Evans' outdoor cross-staff measure of the longitude of the star λSgr by its elongation from a 1981 lunar eclipse, the huge 2°/3 error of which was reported in 1987 in the ever-esteamed Journal for the History of Astronomy (where he's now Associate Editor) as vitiating the JHA-hated R.Newton's claim that Ptolemy's big errors support the JHA-hated case that he faked. DR found that Evans had simply applied parallax (1°/3) with the wrong sign.
This led to the thought that the theory might be fruitful for Greek astronomy. So DR applied it to Hipparchos' two discrepant observations of Spica: was their problem due to his mis-signing his parallax correction each time he measured Spica's elongation from the Moon during lunar eclipses? Yes, again the reported errors dropped from ordmag a degree to ordmag 1'. Details: DIO 16 [2009] ‡1 [pp.3-10].
Later, DIO 16
[2009] (‡1 [pp.3-10]) tested the theory against Hipparchos' Regulus
position and found it consistent with his making the same mistake for
that star, using his observation of the −140/1/27-28 lunar eclipse.
The result agrees precisely with the grossly erroneous attested Hipparchos
longitude (error −35' at his Catalog's −126.28 epoch —
his worst-placed fundamental star).
Thus, the same simple theory solved four separate cases:
Evans, H-Spica 1, H-Spica 2, H-Regulus.
Can anyone name a theory in ancient astronomy star-position studies that
can match such a success?
Though previous investigators had concluded that Pytheas' Marseilles S.Solst ZD (c.300BC) was roughly right, DR tested the consequences of its being exactly right, to its precision. The result places Pytheas in the south suburbs of the city (DIO 16 [2009] ‡2 §B [p.15]), felicitously at a location with an astronomer's ideal sea-horizon due southward (like those of Eudoxos, Hipparchos, & Tycho), far better than Marseilles proper, which looks westward over the water.
Another supposedly rough ancient geographical report was that the latitude-gap from Alexandria to Syene was the same 5000 stades as the gap from there to Meroë. When DR checked the actual latitudes, it was found that this is instead yet further evidence for high-precision Greek work. The intervals are each 7°1/8 or 5000 stades. (DIO 16 [2009] ‡3 §C [pp.22-23].)
DIO 16 [2009] ‡1 [pp.3-10] found that Hipparchos' 3 eclipse-observations were each infected with a sign-slip in the parallax-correction, this had the extra benefit of proving that parallax tables existed in the 2nd century BC.
Further benefit (ibid §D [pp.7-8]): since such tables require sph trig, this discovery provided extra proof that sph trig was being used in the same era. See 1st full bringing-together of such proofs elsewhere on this site — a compilation which naturally includes our remarkable next item:
DIO 4.1 [1994]
(‡3 §D [p.39]) showed that the southern section of
the Ancient Star Catalog exhibited random fractional degree endings
(not the usual plethora of whole degrees one finds in directly visual
ancient astronomical work), a novel mystery that was solved by realization
that the observatons were originally in equatorial coordinates,
transformed via sph trig into ecliptical coordinates.