[One of the characteristics that mark those at the top of their field — those whose self-confidence is founded on genuine total expertise and a realistically grounded sense of secure success — is a warm generosity towards the creativity of others, a gladness in the success of those who might superficially appear to be their competitors. E.g., Mahler towards Strauss. Or van der Waerden to even his most bigoted critics. It is those who inwardly know the infirmity of their views & worth who fear to rise beyond the amniotic warmth of clinging to the numerical & fiscal security-blankets provided by cults of their kind.]
In recent years, certain deliberately-prestigious, captive,
Muffia-servile journals have published
effectively unrefereed history-of-astronomy
research papers, that attempt to find the reliability of data-fits for
two-unknown Gaussian cases. E.g., Centaurus (Y.Maeyama 1984)
and the Journal for the History of Astronomy
(A.Jones 2002 & B.Schaefer 2005).
[DIO has occasional experience in this area. Our non-metric photogrammetry on R.Peary's 1909 “N.Pole” photos was an analytic least-squares simultaneous solution for 22 unknowns: DIO 1.1  ‡4 [p.29].]
Though DIO appreciates (e.g., DIO 6  ‡3 §B3 [p.37]) the valid contributions of these and other opposition scholars, it is a regrettable fact that none of the above-mentioned three authors has understood how to solve Gaussian bivariate problems analytically — or, indeed, by any means besides trial&error or guesswork. Likewise for the 1975 attempt by ultimo-Muffioso O.Neugebauer to solve the Hipparchos-Strabo “klimata” data (also amateurishly investigated by the above-cited JHA 2002 paper) — a politically & ethically scandalous affair (persistent clique-robbery of credit from a great philologist). So we will (below) use the modern klimata controversy as a useful mathematical illustration of how to apply bivariate statistics with ease.
“Klimata” were used by ancient astrologers for house-division (DIO 14  ‡3 n.25 [p.35]). They defined a klima as a latitude L where the longest day M was some discrete value. Ancient klimata tables (e.g., Almajest 2.6) were usually computed at convenient intervals, such as every quarter-hour of M.
It is disappointing that — in addition to their pseudoreferee-undetected technical shortcomings — all three papers appear to be invalidly taking credit for others' discoveries.
This unfortunate history will be detailed in the paragraphs immediately following, but those interested in math more than in our brief passing diversion into academic-clique syciology, may simply CLICK their way past it. (Or to a politics-frei version of the present paper.)
Maeyama 1984's correct but clumsily
induced date for Aristyllos (c.260 BC) was actually discovered
analytically and prominently published earlier by
uncited Rawlins Isis 73 pp.259-265 (1982) p.263.
(Some amusing statistical problems with the 1984 paper are described at
DIO 1.2 
Schaefer's by-now-notorious 2005 paper was — ere mirthful April Fool's Day collapse — touted as the 1st (convincing-to-archons) proof of Hipparchos' authorship of the Ancient Star Catalog; this, despite the previous existence (since Tycho) of multiple reliable proofs of the point, to the near-unanimous satisfaction of 4 centuries of astronomers.
Aubrey Diller's greatest discovery was establishing 2nd century BC use of sph trig, by showing a sph trig function's neat fit to a given set of (now) THIRTEEN (Hipparchos-Strabo) klimata data. Jones 2002 says nothing in favor of said discovery; but he seems to believe (p.17) — on the basis of a non-jelling (DIO 11.1  [p.26]) and literally baseless (DIO 5  n.25 [p.9]) fit to ONE utterly-misunderstood datum — that he has established it himself. (Full details on baselessness will appear if necessary in upcoming DIO 16  §F.) And the DIO 4.2  (p.56 Table 1) proof that including normal ancient 5' rounding ensures Diller's victory — producing a nearly (now fully, as of 2009/4/1) PERFECT fit to all 13 data — is not even cited. H.Thurston & DR were shocked at this — but direct 2002 contact proved fruitless. Nonetheless, hoping for Jones' refereeing-input and his responsiveness to evidence, DIO promptly informed him by email that DR's 2009/4/1 fresh look at Strabo 2.5.36 had found that one of the only 2 reasonable interpretations of this Strabo passage had brought the sole once-apparently-nonfitting datum into precise accord with Diller-DR. (DIO's email included the text of DIO 5  n.25 [p.9] — entire issue still revisable since it and vol.16 are not yet mailed out even as of 2009 early November.) Evidently DIO's attitude of welcoming skeptical checkups (by, e.g., A.Jones & D.Duke) of DR's work — and our thankfulness for the resultant corrections of our (very occasional) errors — is not exactly contagious in opposition quarters. (See, e.g., our grateful, congratulatory front-cover DIO 11.2  delight, welcoming the latest wonderful Jones-Duke discovery, this one undoing an unhistorical DR view.) Instead, in response to the 2009 April email, Jones kept denigrating Diller's find and Strabo's numerical accuracy, somehow missing an unmissably-glaring major advance here, namely, that Strabo's accuracy is now spectacularly & 13-fold-perfectly redeemed. The solution has been revealed not by guess but by mathematical analysis. (Similarly: the DIO math-test that discovered the rocks in JHA Assoc.Ed. J.Evans' head.) And this in a case whose perfect solution is far, far more fragile than hitherto understood. See DIO 16 , e.g., Fig.1 & caption [p.24]. (Published at end of 2009.)
To set all this in perspective: Jones 2002 p.17 explains-away the non-fit of his SOLE attempted match as due to trigonometric “imprecisions”, evidently seeing no significance — not to mention humor — in such patent dodgeball, when competing with a SIMPLER theory that scores over a DOZEN matches without any recourse to such convenient excuses for non-match.
An odd false-appearances scenario (of a sort we realize is required in more familiar theologies). It only looks like DIO is right and JHA wrong; but that's just an illusion — something akin to the devil's work. A tightly-focused maleficent force must have sullied what would-surely-have-been a perfect match for Jones' theory — and (to spoof Jones 2002 n.9) sly satan even more fiendishly polluted the Diller-Neugebauer-DR klimata-data-base in just such a way (by adding a secret 100 stade constant onto all klimata) as to obscure the now-buried truth (which 7y of subsequent research has failed to recover, despite the stimulus right along of a DIO van der Waerden Award for anyone who finds a better solution here than Diller-DR's for the klimata) while creating a false data-set that happens to hit the bulls-eye: a target MUCH more microscopic than JHA's archons knew when gleefully publishing the 2002 fantasy. [Ever heard of an elephant challenging a mouse to a pistol-duel? — or, better, a dueling elephant who imagined his mouse-opponent was as big as an elephant?] Thus creating-by-pure-wild-flukish-luck a PERFECT fit to a sph trig model based upon [i] satan's constant, and [ii] the only accurate obliquity we (by independent verification: DIO 5  §D3 [pp.8-9]) know was anciently used and was Hipparchan.
Let's be clear about this: there is no harm and much benefit in attacking by looking for an alternate theory, especially when entertaining in-passing such an imaginatively exploratory one as we see in Jones 2002 (though its untenability should have been obvious long before publication). Indeed, Jones & Duke assisted DR magnificently, by finding a valid alternate theory, displacing an invalid 1987 DR theory. The problem here is the failure to own up when it is one's own attack that is invalid — and in this case assists in postponing immortal credit to a highly eminent fellow philologist.
Since Jones 2002 could show zero predictive hits (vs a flock for Diller), what was the point of publishing it? The paper too closely resembles classic Muffia try-anything sand-in-eyes harassment (of non-Muffia scholarship) to permit our avoidance of such an otherwise-uncomfortable question.]
Though having written his above-cited prominent 2002 paper on
the Diller matter, Jones now (following DIO's
email) made no attempt to converse
with DR, and just (2009/4/9) brushed off
the whole issue as minor.
A particularly ironic contention:
[a] It is fruitless to gratuitously insult the import of Diller's obviously major contribution to the history of mathematics.
(Has DR's involvement become the insuperable impediment? Has once-seemingly neutral Jones adopted the path mapped out at DIO 2.3  ‡6 n.14 [p.93]? See also ibid §E1 [pp.93-94].)
[b] The issue was unminor enough for Jones to have volunteered as the sole scholar in 1/3 of a century (1975-2009) publicly to attack Diller's theory, on what he (incredibly) regarded as substantial grounds, rushing this into JHA print at the very moment when Isis was seeking refereeing (from, e.g., hist.astron mogul and DR's indefatigable denigrator O.Gingerich of JHA) for the eventual 1st-ever hist.sci-journal citation (H.Thurston Isis 93.1:58-69  p.67 n.18) of the very DIO 4.2  Table 1 [p.56] which the Muffia cult had hitherto successfully pretended didn't exist. Jones' paper continued this shun-sham anyway, during its failed try at finding the grail of a plausible Alternate Explanation to Diller-DR's.
As Thurston's Isis paper was circulating in 2001, several scholars finally began (note parallel to a prior similarly-timed Muffia tactical switch from omertà to counter-attack: DIO 2.3  §E3 [pp.93-94]; and DIO 4.3  ‡15 §§D6&F4 [pp.126-127&130]) trying to-the-max to puncture every one of the numerous DR discoveries cited in the Isis paper. These probes (which should have been forthcoming collegially years earlier — not instead as a seething last-ditch effort to squelch hated heresy) fortunately resulted in circulated attacks on DR's proposed solutions for the Almajest Mars-Jup mean motions and for the Hipparchos klimata. The former attack (Jones-Duke) was valid, the latter invalid. (Note that the fact that DR's many other discoveries survived all 2002 attacks hasn't since enlightened or un-seethed a single Muffioso. I.e., if a math-check were negative, that would be reported. If positive, nothing is said. Needless to say, that is not DIO's approach to its opposites.)]
Desperate-Institutional Ethics Where the Shun Don't Shine:
The Jones 2002 paper under discussion appeared in the JHA of pol's-pol Gingerich, whose genius is such that, even as he nears age 80y, he can super-psychically, with absolute certainty (eternally self-circumscribed by inescapable shunner-improvidence: DIO 6  ‡1 §J3 [p.25]) perceive that every single contribution in DR's lifetime of discovery is worthless — since each
have an Alternate Explanation.
Including even all future DR discoveries. Sound incredibly weird? Yes. But consider: how else can a shunning be made to stick? Especially when the repulsive art of shunning is so deeply dark that its chief designers modestly will not even (publicly) take credit for its martially-strict maintenance.
A Chimeral Castle-in-the-Air & a Trip From Sane to Zane:
J.Evans' 1987 JHA monster-length double-Pb papers provided yet another link in the decades-long chain of an ever-subsidized tradition of shots at making-real that alluring castle-of-eliminations-of-every-DR-result that has hovered tantalizingly for 1/3 of a century in the shunny air between frustrated OG's ears — a project which Jones alone has ever been able to lay a single actual history-of-astronomy brick of.
Two Evans-vs-Jones distinctions worth noting:
[i] Encouraged in 1994 by non-cultists (the highly able scholars Curtis Wilson & Hugh Thurston), Jones formerly went outside Muffia bounds in assenting to a few DR discoveries; but, as those benign influences gave way to Muffia ones, he appears to have become priority-hypnotized out of the former passing phase.
[ii] Jones 2002 delusionally-aggressively tried (p.17) to replace sane theory with zane theory. (This in an instance in which the two theories were not even mutually exclusive: see 2002-2003 printing of DIO 4.2  inside front cover n.2 [p.54].) Said folly is beyond even Gingerich's slightly less obvious standard-antiDR gambit, which is merely to create Alternate-Theory doubt (the only recourse, when as always DR's math maddeningly can't be overturned) — not to defy common sense by prosecuting his shaky Alternates as exclusively true. (An example from a different controversy: by simply proposing that all sorts of credibility-stretching theories [other than DR's obvious one] might-might-might be true, Evans 1987 followed his mentor Gingerich's recipe by taking care not to come down in favor of one of his joke Alternates, thus he never committed himself by going so unjustifiably far-out as Jones 2002 has.)
[Another distinction: when Britton (c.2000) told DR he was siding with Neugebauer's authority vs Diller's math it was at least another's authority. Jones' 2009/4/9 brush-off of Diller-DR is based on Jones' idea of the ultimo Authority: Alexander Jones. (Jones 2002 p.17, or DIO 16  ‡3 §G3 [p.31]: “I believe we have to regard the shadow ratio [(41 4/5)/120] as the more trustworthy datum”.) Jones' self-confidence is sometimes a plus. And DIO is the better for that. But this is not one of those instances.)]
Fast-forwarding from 2002: cultism's then-8y of nonciting Diller-DR's 1994 table has by 2009 become virtually doubled to 15y.
And nowadays, we are fortunate to be able to announce a miracle of cultist freewill-death&transfiguration: thanks to the JHA's inimitable archonal trio, the seemingly-aging Muffia has grafted-devolved into a hybrid hydra — as mentally-deadicated as ever to its unthinkingly rigid idea-killing exilings — which (with due respect to its somnambulistically obedient immortal-zombiedum) we may dub double-entrendrily:
[We will continue using the older term in most cases following the present document, for several reasons too obvious to belabor here.]
The very reason why the above-cited 1994 table has been Muffia-Morphia-shunned for 15y IS its embarrassing UNminorness for the history-of-ancient-astronomy clique; the table's virtually (now exactly) perfect fit showed more simply, dramatically, embarrassingly, & consistently than any other case (though Ptolemy's Venus double-date Experimentum Crookis comes close: DIO 11.3  p.70) how densely and-or politically evidence-immune said clique has been. As noted during the 1994 announcement of Diller's vindication through accounting for ancient-standard 5'-rounding (see DIO 4.2 pp.55-57, Table 1), even young children perceive his case's coherence. (Their secret? Genuine neutrality. Which, revealingly, turns out to be much more vital than Muffia semi-numeracy for discerning the truth of the issue.)
From DIO 11.3  ‡6 n.12 [p.73]: the Muffia cult keeps insisting (ibid n.13 [p.73] & DIO 1.1  ‡3 §D [p.20]) “that the [Ptolemy Controversy]'s losers are the true experts, and the winners are mere cranks … all this, while themselves convincingly imitating the key feature of cranks: clinging to long-held opinions despite avalanche after avalanche of evidence against them.”
On 2009/9/30 DR left two messages on Jones' phone-mail, asking that we chat,
emphasizing hopes of producing amicable agreement on the klimata issue
and wishing that he would consider at last publicly acknowledging the merit
and strength of Diller's original discovery.
(The fruit of a scholar as creative as himself.
Given Jones' deserved eminence, his recognition of this would
[outside the eternally ineducable: Old Guard Muffiosi
& aHoly Trinity atop the JHA]
end the Diller-Strabo pseudo-controversy at a stroke
— and [for this history] an atypically amiable stroke.)
Jones was simultaneously informed of the uploading of an early version of the present document (the linking of which was postponed until 2009/11/3), which delineates the statistical situation of Diller-vs-Jones.
Jones emailed DIO that he'll be busy for some weeks.
Disappointing so far. Anyone can make mistakes. (DR certainly does!) But persistent failure to acknowledge them goes beyond mere error (especially in the context of ALL associates' like rigidity). It ultimately disallows blaming others (pseudo-refs in this case) and concentrates attention on the perp's character — as regards, e.g., humility, self-confidence, & etc. I.e., it is questionable whether the 2002 paper can much longer merit even a tenuous defense painting it as just a mistake.
(Was it ever credibly thus evaluable? See DIO 11.3  ‡6 n.20 [p.75]. Or did it constitute a made-man cult-initiation requirement? Does academic advancement in some fields actually require going counter to academic principles?)
Given the cited journals' problems
with bivariate statistical analysis,
it may be a mercy to lay out a fresh, direct method that requires
no knowledge of how to compute a least-squares problem analytically.
And, some good news for the uninitiated:
computing a two-unknown probability P
is in some respects easier than for any other number of unknowns.
[It helps that when computing probability P from probability density pd for even numbers of unknowns, we do not need to use the difficult error-function. Here, throughout, we compute P as the cumulative probability exterior to the locus of points with the same pd as the point of interest — exactly as is routinely done for 1-dimensional statistical analysis.]
Most textbooks give a formula for the bivariate probability density pd that is not only messy but requires specialized analytic investigation to determine the inputs: the two unknowns' standard deviations and their correlation. So some may find it a pleasant surprise to learn of a clear, accurate, & efficient method for finding P while avoiding all that bother.
We have a set of N values of a variable j for specified values of an independent variable t, and there is a proposed function f(x,y;t) that will fit the values of j as closely as possible — for the best choices of two unknowns: x&y.
We compute function f 's value corresponding to each of the N values of t for which values of j are available. Each difference between a datum j and the computed function f(x,y;t) [which is supposed ideally to equal j] is called a “residual” Δ, with sign-convention such that Δ = j − f. The probability of any given pair of values of x&y is measured by the sum S of the residuals-squared:
At the x&y where S is minimized (values which could be found by trial without sophisticated analysis, as already noted), a subscript m is appended to x&y and to S.
Our quick&undirty exact solution-for-amateurs will now be set up by a trivial calculation, finding the normalized or Relative Difference D between sum S at the point (x,y) of interest versus that at the minimum:
Then our solution for P is simply:
where (again) N is just the number of data.
That's all. It's that simple. And it's as accurate as any other method.
[It should be noted that all standard procedures must find the foregoing quantities (S & its minimum, etc) anyway, en route to their more elaborate solutions — e.g., finding standard deviations σ, correlation(s), pd, and so on. So if finding P is the goal (and it is obviously the main one), our equation makes P easily accessible at the outset, obviating any necessity for getting into complexities beyond. (The main advantage of sophisticated least-squares analysis is that it will discover xm, ym, Sm, P, and all σs & ρs without trial&error's tedium.)]
An example always helps. So let us examine the above-cited
infamous case of the
data which are discussed and explanatorially tabulated in
DIO 4.2 
[pp.55-57] (Table 1);
DIO 5 
(Table 0 [p.7], seven Diller-predictive-success
confirmatory items 1-7 [pp.8-9], n.25 [p.9]); and
DIO 16 
‡3 (Tables 1&2 [pp.20-21]).
These data Aubrey Diller discovered in 1934
(Klio 27:258-269  p.267) were anciently
calculated via sph trig
for obliquity ε = 23°2/3.
No modern scholar or extant ancient ms had previously recognized either  this obliquity or  the method by which Strabo's data had been calculated. (Diller's late-Hipparchan obliquity 23°2/3 is the most accurate value known to have been used by any ancient astronomer.)
Predictivity details: Since Diller 1934, two new Hipparchos-Strabo klimata have surfaced. Both are in EXACT accord with Diller's sph trig formula. During the same 75y period, several independent evidences for obliquity 23°2/3 have appeared, adding yet further predictivity-successes to the Diller scorecard. No other theory has produced any.
Yet, in one of the three statistically inelegant papers cited
earlier here, A.Jones
— to whose knowledge & creativity DR has substantial debts (e.g.,
DIO 11.1 
‡3 §§D1-2 [pp.22-23], and
DIO 11.2 
DIO 14 
‡3 §I2 [p.44]) —
(JHA 33:15-19 )
the following typically original theories:
[a] that Diller's obliquity ε should have been not 23°2/3 but 23°51'20", and
[b] that a hitherto-undetected constant A might have been added in antiquity to all Hipparchan klimata.
In the following response to this strange proposal, we will cooperatively treat ε as an unknown and (to test Alexander Jones' shift-speculation) add another unknown A to the anciently standard sph trig equation for latitude L (in degrees) as a function of longest-day M (in hours), thus:
Comparing this function to the Hipparchos-Strabo klimata data: we seek the best-fit choice of ε&A, as well as a way of gauging the probability of all other ε&A pairs. Here, L, M, ε, & A take the respective rôles of j, t, x, & y in our earlier explanatory paragraph. The data L are the latitude values given by Strabo, for 13 distinct klimata M, ranging from 12h3/4 to 19h.
A table is provided (below) of the Hipparchos-Strabo klimata latitude L
data (in stades, consistently reported by Strabo in hundreds of stades),
adjacent to competing theories' computed L
(rounded here to 1 stade) and their residuals Δ,
with sign-convention Δ = Hipparchos-minus-calculation.
The last rows of the table display residual-sum S, normalized r (see below), and probability P:
|Klima||M||Hipp L||best-fit L||best-fit Δ||Diller L||Diller Δ||DIO L||DIO Δ||JHA L||JHA Δ||Princttt L||Princttt Δ|
One easily sees that the column which perfectly matches
Strabo's figures (to the 100-stade precision of his reportage) is
the DIO one, which realizes (e.g.,
DIO 4.2 
p.55 n.6) that each L in degrees had previously
(before conversion to stades at 700 /degree) been
to the nearest 1°/12 (5'), according to ancient geographical convention
for klimata (Geographical Directory 1.23).
[The ancient calculator appears to have achieved extreme precision. (Quite a revelation, considering that we are discussing trig tables virtually at what has over-confidently been regarded as the time of trigonometry's invention.) When carrying out the scheme Diller discovered: an L error of merely 1" could have ended up misplacing the 18h klima by 100 stades, since application of the sph trig equation for M = 18h produces L = 58°12'31". (Or 40746 stades raw, vs 5'-rounding to 58°1/4 = 40775 stades or 40800 stades by DIO's theory. See both entries in the 18h row of the table.) The residuals of the DIO solution are not converted into probability since (as the table shows) their S is actually 25% smaller than the best-fit Sm for unrounded L data!
(Choosing 5' for rounding-precision, might be regarded as a 2nd unknown for the DIO solution; but this is an attested standard ancient value, so actually it's a known.)
At the other extreme, the grotesquely outsized S of the long-sancrosanct Princetitute scheme occurs despite Neugebauer's manipulation of a repeater-complex of four unknowns (vs Dillers' one — thus giving Diller's theory a clear Occamite preferability). The four unknowns are the coefficients of his proposed cubic-polynomial solution, L = 50(M3 − 62M2 + 1307M − 8454). (Computations for the Princetitute theory should use F = 13 − 4 = 9, not 12, as for Diller's 1-unknown theory; nor 11, as for Jones' 2-unknown theorizing. But, when comparing, keep in mind that the higher-dimension normal hyper-surface for fitting the cubic is nothing like that for the Diller sph trig solution.)
If we solve by least squares [which was perhaps beyond Neugebauer's experience] to find the 4 best-fit cubic-polynomial coefficients, we find that (though far better than the Princetitute cubic) even with its 4 unknowns it leaves a σ (32 stades) about the same as that of the 2-unknown Diller-Jones function and slightly inferior to that of a 1-unknown fit to Diller's pure sph trig function (31 stades) Going quartic does not improve matters due to F 's shrinking from 9 to 8.) But of course resorting to ever-higher order polynomials can tighten σ more than the lower orders.
This is because of a crucial point that will make clear to any experienced mathematician that the Diller-DR final theory is valid: the reason that high-power polynomials (quintic or higher) are needed to get σ below 30 stades is that sufficient numbers of unknowns allow some threading through the local roughnesses caused by roundings, since no smooth function can overcome the kinks in the data-curve that correspond to the two anciently-conventional roundings (100 stades and 5') that inject ripples into the pure sph trig equation's curve. These kinks (deviations from smoothness) are about 0°.1 at 2 rough points: [a] From M = 14h1/4 to M = 14h1/2, and [b] from M = 16h to M = 18h. Each kink is primarily due to one of the two klimata which failed for the original Diller scheme (but was salvaged by the introduction of accounting for ordinary ancient 5' rounding, which Diller 1934 suspected was a potential source of non-fits): M = 14h1/4 (DIO 4.2 ) n.10 [pp.56-57]) and 18h. From 6-unknowns (quintic polynomial: σ = 11 stades) on up [none of which, as Keith Pickering notes, has anything to do with how ancients would solve the klimata problem!], σ is suddenly much tighter. I.e., it requires a six-unknown polynomial (to match the data's general curve and additionally snake through its rounding-kinks) in order to exceed Diller's plain 1-unknown solution. (A 6th order polynomial fits all klimata within 2 stades. By the time we get to a 12th-power polynomial with 13 unknowns, we of course get a perfect fit, but σ will be indeterminate since the equation for it will have null numerator & denominator, and — having thus eliminated over-determination — we have made the problem a no-longer-statistical one.)]
I.e., a philologist chose a better function (to fit to the klimata data) than did an eminent Ivy League & Princetitute mathematician, the Muffia's ultimo-gooroo: O.Neugebauer. (Who multiplied his initial error by not listening to the wiser chooser — and then abusing [and branding as a fool] the latter scholar — for the offense of disagreement.) One can see why admitting the folly and the offense has been harder for the Muffia's Ivy Leaguers & Princetitooters than submitting to torture. (Unless torture entails debating DR.)
NB: The Neugebauer-Princetitute cubic-polynomial theory was promoted for over 2/3 of a century as the highest wisdom on the matter entirely because it ascribed the klimata to Babylonianesque math (Neugebauer HAMA 1975 pp.305-306 & 334; earlier: Exact Sciences in Antiquity 2nd Ed. BrownU 1957 pp.157&176), though nothing in Babylon's voluminous extant records mentions klimata. (See, e.g., ibid p.158.) Indeed, Babylon's astrologer-priests didn't even have plane trig, much less the sph trig required to compute klimata.
From the numbers in the table, let's get an idea of just how outré the Princetitute solution is. If we artificially compare this 4-unknown solution's S (4284816 arcmin squared) to the 2-unknown minimum (11279 arcmin squared), we have D = 4284816/11279 − 1 ≈ 379. So the artificial normalized probability density is:
In other words, if we use the probablity density at the 2-unknown minimum-point as a standard (though it corresponds to a solution that is distinctly inferior to DIO's), the probability density for Neugebauer's Babylonianesque scheme is smaller by a factor of 1 followed by 740 zeroes. That's:
A more generous and less hybrid approach would not ask the lavish 4-unknown
Princetitute cubic polynomial solution to match Diller-DR's sph trig
sharp-shot Derringer —
but instead just compare it to the best of its own type & no. of unknowns.
As already noted,
the Princetitute S = 4284816 arcmin squared.
However, the cubic polynomial that best fits the 13 klimata data reduces the residual-square sum (in arcmin squared) to Sm = 15995. So
And we know that F = 13 − 4, thus
[Including our elsewhere-proposed 14th klima (the Equator: = 12h) will lower these odds but will simultaneously destroy 13 out of 14 fits: all but Phoenicia (M = 14h1/4) whose residual would be just +33 stades, thus ranking as a fit, given Strabo's 100-stade rounding of all klimata L. Meanwhile, the Diller-DR theory fits all 14 L data.]
After treatment analogous to procedure described below
(translation, transformation, & normalization
but here in 4 dimensions), we have an isotropic distribution,
where the hyper-spherical locus of all points with Princetitute S
has normalized radius r, with
r equalling the square root of F·D.
Integrating over all such hyper-spherical-shell loci from there to infinity, we find
A general asymptotic rule, for any number U of unknowns, can be formulated: for large r,
So the best we can do for Neugebauer's cubic polynomial's odds of
being valid — even in the limited context of other potential
cubic-polynomial solutions — is
(applying the above 4-dimensional formula
for P): 1 followed by merely 518 zeroes.
[When R.Newton computed high odds against the legitimacy of Ptolemy's “observations”, Muffiosi just scoffed that you can prove anything with statistics. (Similar to what's discussed at DIO 16  ‡1 n.8 [p.5].) Which tells us that: [i] Most such sour-gripes critics don't even understand the math. [ii] They are innocent of the fact that lying-with-stats is done not by math-trickery but by disobeying the foundations of the math. (Note also that the present case differs in that the data are calculations [not alleged observations] and thus less subject to unruliness.)]
Is 13-Out-of-13 Enuffia?:
Again: FOR DECADES, the foregoing joke of a solution was Muffia-Princetitute holy writ, dissentlessly worshipped by the Muffia (which elevated it even into the Dictionary of Scientific Biography) without a peep of dissent, though its 4-unknown juggling could satisfy only 6 out of 13 data — all the while exiling the Diller-DR solution which, by (Diller's mathematically-derived) selection of a single unknown, satisfied all 13. Again: it's the Morphia that routinely classifies other scholars as cranks.
In a 1997 addendum, DR commented
(DIO 4.2 
[p.57]): “Muffia peddlers of funny
explanations for pre-Ptolemy Greek astronomy keep non-citing
(i.e., faking the nonexistence of)
our stark Table 1” [ibid p.56
or here at above] — and, 15y later,
those subtle-as-ever darlings are still staunchly maintaining
the same immutable demonstration of their collective integrity.
(Morphiosi damn non-citation as execrable scholarship when attacking heretics,
themselves engaging in decades of deliberate
For eyepopping details of such hypocrisy at the top of the JHA,
see DIO 8 
inside front cover [p.2; especially n.5].)
That there might be occasional individual scholars who fall short here is not the objection. The scandal is that EVERY single member of an entire politically well-placed, dissent-choking cult has — zombie-hypnotized by careerist priorities — kept Muff-loyally noncite-shunning the 1994 table (DIO 4.2  p.56, reproduced above) a table which ironclad-vindicates a great scholar's long-abused, once-heretical, but patently valid discovery.
For 3/4 of a century. And counting.
By not bailing out when contra-Muffia evidence loomed (note common-sense at DIO 10  endnote 21 [p.105]), the Morphia is now in-so-deep as to allow no chance of pursuing any strategy other than Martingalesque stakes-redoubling via standard Alternate-Theory-bluff, while trying to prevent the public from knowing of the only theory that fits. Hmmm. Parallelling a self-evident query (DIO 9.3  ‡6 §C7 [p.122]) which DR has put to a different cornered cult, let's ask:
if the table isn't convincing, why such fear of citing it?)
Back to reality:
Analysis or testing on our table's data finds S is minimized at ε = 23°37'.60 (primes signify arcmin = sixtieths of degrees) and A = −2'.44 (= −28 stades, at the Eratosthenes-Hipparchos-Almajest scale of 1° = 700 stades); minimum Sm = 11279 square stades or 82.86 square arcmin.
As noted above,
Jones (JHA 33:15-19 )
argues that Diller's values and data are both invalid, contending that
ε ought to be 23°51'20" (the Eratosthenes-Ptolemy value) and that
A looks like it ought to be +100 stades or 8'.57.
Yet Another Muffia-Discomfitting Surprise: a Hitherto-Unnoticed 14th Klima.
(We do not include it in our stats since Strabo 2.5.34 says he won't deal with the Equator region.)
Jones appears not to have noticed that his 100 stades scheme implies that Strabo's allegedly-corrupt source was claiming that the 12h klima (i.e., where the days & nights are equal) is 100 stades north of the Equator! (And Neugebauer's scheme has the 12h klima 1500 stades — over 2° — north of the Equator….)
[Moving the equinoctial L to +100 stades isn't Jones 2002's only imbalance: the paper gives Diller no explicit credit for his remarkable pioneering discovery that the Hipparchos-Strabo data were computed by sph trig — thus for the 1st time firmly establishing that branch of mathematics' existence to the 2nd century BC. Yet Jones implicitly assents to this central truth, when he says (ibid p.17) “one may well suspect that one or two modest changes in the intervals, through either scribal error or deliberate tampering, could have introduced systematic errors which would affect the value of the obliquity best fitting the data.” But Jones approaches the problem of solving for the speculated tampering by adducing Hipparchan data exterior to the Strabo data-set, rather than searching by math analysis, as we do here.]
We now will test Jones' JHA proposal by our compact least-squares method.
For each Strabo M, one compares the computed L (found by substituting Jones' ε&A into the above formula) to the L given by Strabo. (See table.) The residuals Δ (Hipparchos L − computed L) are each squared and the sum of those Δ-squares is formed. This sum is: S = 539.8 square arcmin.
Our simple method computes as follows: relative diff D = 539.8/82.86 − 1 = 5.598; multiply D by 11 (since N − 2 = 13 − 2 = 11); divide by 2; & invert the natural anti-log:
For the JHA proposal (ε = 23°51'20" & A = +100 stades), we find:
On the other hand, for Diller's values (ε = 23°2/3 & A = 0), we find S = 88.78 square arcmin, so D = 0.0718, and
(Both values are of course found in
the P row of our table.)
Given the contrast
between the P, there is no need for subtle analysis to choose between
the Diller-DIO & the JHA solutions.
[Accounting for a natural human impulse to seek outs, it is necessary to add here that resorting to insisting upon formerly-orthodox 11800 stades as the 13h klima's L (instead of DIO's obviously valid 11600 stades) will not salvage MuffThink, though the odds are smaller. (Since 11800 is glaringly discordant with all contending theories, experienced statisticians would have discarded it even before the 2009/4/1 vindication of accordant 11600.) If one nonetheless stubbornly insists on computing with 11800 (thereby ludicrously super-inflating σ to 5'.0), Diller's solution is still statistically acceptable at P ≈ 0.22, while Jones' is still severely non: P ≈ 1/30000.]
NB: Our table shows that the DIO fit's tightness actually exceeds even that of the least-squares-test's best-fit — and by a considerable margin, S being less by 25%. This dramatizes how spectacularly precise is the fit effected (to a problem of extraordinary sensitivity), by DIO's introduction of standard ancient geographical rounding of angles to 5' precision.
We now temporarily return to general analysis (which will ultimately show the equivalence of the simply & the elaborately obtained values for P).
A contraction (followed by a generalization) of our simple method can be effected through the standard definition of “degrees of freedom”:
where N = the number of data, and U = the number of unknowns (2 in the bivariate case, by definition). Thus, the earlier equation is seen to be but one instance of the more general rule:
[One of the advantages of this rendition of our solution is that while it provides P only for the 2-unknown case, it supplies pdn (the normalized pd) for any number of unknowns. (Normalization in this application refers to re-scaling pd such that it equals unity at the minimum point.) However, keep in mind that P, the true measure of a point's probability, is the exterior volume under the pd surface — just as for the 1-unknown problem, the area under the normal curve's tails (likewise the region exterior to all [both] points of the same pd) is the true measure of a point's likelihood for that case. (Which is why the common 1-unknown problem's P is expressed with the error-function, instead of the much simpler function we've seen will express pd. The error-function is not analytically integrable, so it is customarily dealt with via series approximations — or tables pre-computed therefrom.)]
Despite its brevity, our simple equation
is not an approximation.
[Except insofar as most such problems only stay virtually Gaussian near the minimum point. But that warning applies equally to the sophisticated standard methods we will glance at below.]
The usual approach to bivariate analysis sets up an x-y-z coordinate frame where x & y are the unknowns, and the z-axis is for the probability density pd, which is an elliptical-cross-section Gaussian function (“normal surface”) on the x-y plane. In x-y space, the point providing the best fit (maximum pd, minimum S) is at x = xm & y = ym,
We will call the standard deviation of a single datum σ, which is:
In the case of the ancient klimata, the equation gives σ = 2'.74 = 32.0 stades for testing Jones' 2-unknown theory.
However, note that, though (actually somewhat because) Diller used but one unknown, the same equation for σ shows he has a tighter solution with his 1-unknown theory: σ = 2'.72 = 31.7 stades. (Less than the 32.0 stades yielded by 2-unknown analysis, since the 1-unknown computation of σ uses a bigger F: 13 − 1 = 12, instead of the F = 13 − 2 = 11 which we have used for 2 unknowns. The best 1-unknown least-squares fit is at ε = 23°39'.61±0'.62. Resort to the error function shows that (for normdev = [23°2/3 − 23°39'.61]/0'.62 = 0.63), 23°2/3's two-tailed probability P = 0.53. So from either 1-unknown or 2-unknown perspective, the Diller solution is easily compatible with the Hipparchos klimata data.
Returning to 2-unknown analysis:
We signify each unknown's standard deviation by σ subscripted by that unknown. Even when simplified by translation of the x-y origin to the best-fit position, the standard expression is still cumbersome:
where ρ is the correlation of x&y.
But let us try a more elegant approach, which will end up confirming the simple method given earlier, determining P, the sum of all probability on the ε-A plane outside the locus of points (on said plane) whose S and (thus pd) equals that of any ε-A point we wish to investigate (e.g., Jones' ε = 23°50'20" & A = +100 stades).
The excess SS of the sum of the square residuals S above the minimum such sum, gauges the probability density of any point (ε,A). If we normalize the pd to pdn through dividing it by its value at the minimum point, we have:
In the present problem, the number of degrees of freedom F = 13 − 2 = 11; and we know how to find σ.
We note in passing that all this gives for Diller's ε & A (where, as already noted, S = 88.78 square arcmin & σ = 2'.74):
— where we see that our math (for density pdn) obviously parallels that of our simple method's earlier computation of cumulative probability P.
The surface generated by our latest pdn equation is (but for scale) the same as that for pd. Yet both bear the inconvenience that — because x&y are correlated (highly so for our test equation) — their elliptical cross-sections (which are, after all, loci of constant S and thus pd) are not oriented such that the major&minor axes are along the x&y axes (ε&A axes in the Diller-klimata investigation).
But that difficulty can be eliminated by rotating the x-y plane so that the new x'&y' axes are the eigenvectors of the old frame.
For the Strabo-Diller case we've been using as an example,
the required rotation is almost exactly 50°.
The matrix, which relates the unknowns' uncertainties (and correlation)
to σ, is diagonalized by a corresponding similarity transformation.
For the Diller-klimata case, the resulting matrix's diagonal elements are
in the rather dramatic ratio of c.100-to-1,
showing that the relative standard deviations of
the new unknowns (x', y') are in c.10-to-1 ratio.
[The quadratic secular equation is: E2 − 3.3126E + 0.1054 = 0. This produces eigenvalues of 3.280 & 0.03213 (ratio c.100-to-1), which are the diagonal elements of the newly transformed matrix.]
Correlation ρ — so inconveniently high in the former standard equation — is now mercifully zero, because the new normal surface is symmetric about both the x'&y' axes, reducing said equation to:
In passing, we check pd for Jones' proposed ε&A,
where x' & y' are 17'.36 & −3'.441, respectively,
and the corresponding standard deviations are 4'.91 & 0'.491, resp,
so the normalized deviations
are 3.54 & −7.00.
The pre-diagonalization norm-devs are 4.34 & 2.94, which are extremely misleading as to the actual odds here (a recommendation for our method, which is simpler and never misleading); this is due to high correlation: ρ = 0.980 (while in the new frame ρ = 0). Nonetheless, if pdn is computed by normalizing the more complex earlier standard formula, the result is the same as that computed from the formula provided just above. Likewise for Diller, where the pre-rotation normdevs are 0.753 & 0.644, we find pdn:
When the result is divided by the pd computed for the minimum point
(i.e., by 1/2π), this normalized pdn < 1 in 10 trillion,
which predictably agrees with P.
[A precise equality for 2 unknowns, which holds only crudely for other cases: at points far from the minimum-pt, P is about equal to pdn multiplied by the product F·D taken to the N − 2 power. (We ignore a proportionality constant of trivial effect on the exponent — and which cancels out anyway when computing P.)]
Our numerical example reminds us that further simplification of the situation can be effected by general normalization of the unknowns: adjust (divide) each new (primed) unknown by its own standard deviation.
It is obvious from the equation for un-normalized pd that we now have:
This new probability density is a radially isotropic (circular-cross-section) Gaussian surface on the normalized x'-y' plane. We exploit a well-known conversion to polar coordinates r&θ, where
Integrating over θ from 0 to 2π produces a probability density pd ' that is a function only of normalized r:
This is a much easier expression to deal with than the usual Cartesian pd.
For probability P, we integrate from any point's r to ∞, to find the volume (under the pd ' function) exterior to the (circular) locus of points with the same r (and thus pd '); this produces:
which is far simpler than the error function which is unavoidably involved in parallel integration for the familiar case of 1 unknown (or indeed any odd number of unknowns).
For Diller's ε = 23°2/3 & A = 0, where r = 0.889, the same formula gives
Returning to general analysis: using (with our definition of D) the fact that
— the very same compact shortcut earlier proposed here.
 The probability equations involving S are more basic than the derivative ones involving standard deviations.
 The reason our various methods give consistent answers is that P, the proportion of the volume (beneath the normal surface) exterior to the locus of constant S, is invariant under our successive transformations: translation, rotation, normalization-rescaling.
Thus, our compact method is as accurate as any. While far simpler.