Beat MuffEstablishment MultiShooter Blunderbuss Army

GetSmart Asks: Would-You-Belieeeve 14-OUT-OF-14?

A Clear Crucial-Test Which Occam-Razor-Sharpens Our Curiosity:

How Does the Lock-Step History-of-Ancient-Astronomy Community

Differ from Herds of Projective Crank-Accusing Religious Cultists?

TruthShun Priority SleepWalks On:

MUFFIA Devolves Into MORPHIA

Dennis Rawlins 2009 [rev 2018]

[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 and worth who fear to rise beyond the amniotic warmth of clinging to the numerical & fiscal security-blankets provided by cults of their ilk.]

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),
the

[

Though

“Klimata” were used by ancient astrologers for house-division (

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.)

OpenMindedness Atrophed in the Tight

History-of-Ancient-Astronomy Cult?

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

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)

To set all this in perspective: Jones 2002 p.17 explains-away the non-fit of his attempted derivation of the origin of obliquity ε = 23°51'20" as due to trigonometric “imprecisions”, evidently seeing no significance — not to mention humor — in such patent dodgeball, especially when competing with a SIMPLER theory that scores over a DOZEN on-the-nose klimata-fits

An odd false-appearances scenario (of a sort we realize is required in more familiar theologies). It only

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 only helter-skelter predictive hits on klimata (vs all klimata 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.

NB. Jones' over-the-shoulder fleeing of engagement on the klimata issue is not the only evidence that he is consciously engaging in academic misbehavior:

[a] Why is he the only one of the four participants (Diller, Neugebauer, DR, Jones) in the klimata debate who has not published a table of his theory's fit to the data? — which would of course immediately reveal its invalidity.

[b] He has repeatedly non-cited the superior-fit DR edition of Diller's table: originally at

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

[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

As Thurston's

** Desperate-Institutional Ethics Where the Shun Don't Shine**:

The Jones 2002 paper under discussion appeared in the

Including even all

[A creditable exception should be noted: in 2005 (

** A Chimeral Castle-in-the-Air & a Trip From Sane to Zane**:

J.Evans' 1987

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

[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

Fast-forwarding from 2002: cultism's then-8y of nonciting Diller-DR's 1994 table has by 2016 become more than doubled (while heading for trebled [at 24y by 2018].)

And nowadays, we are fortunate to be able to announce a miracle of cultist freewill-death&transfiguration: thanks to the

THE MORPHIA.

[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 24y]

From

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' [formerly] deserved eminence, his recognition of this would
[outside the eternally ineducable: Old Guard Muffiosi
& aHoly Trinity atop the ** JHA**]

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

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

Jones' Gratuitous Attack on Diller

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 start
with a trivial calculation, finding
the unknowns' normalized deviations from their best fitted values,
via the relative Difference *D* between
the sum *S* at the point of interest (*x*,*y*)
vs that (*S*_{m}) at *S*'s minimum (best-fit):

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,
without trial&error's tedium, it will discover
*x*_{m}, *y*_{m}, *S*_{m},
*P*, and all σs & ρs.)]

An example always helps. So let us examine
the above-cited
infamous case of the
Hipparchos-Strabo klimata,
data which are discussed and explanatorially tabulated in
** DIO 4.2** [1994]
[pp.55-57] (Table 1);

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** [2002]
‡3 §§D1-2 [pp.22-23], and

[a] that Diller's obliquity ε should have been not 23°2/3 but 23°51'20", and

[b] that a hitherto-undetected constant

In the following response to this strange proposal, we will cooperatively treat ε as an unknown and (to test

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 | JHAL | ΔJHA |
Princttt L | Princttt Δ |

Cinnamon | 12h3/4 | 8800 | 8812 | −12 | 8824 | −24 | 8808 | −8 | 8848 | −48 | 10202 | −1402 |

Meroë | 13h | 11600 | 11602 | −2 | 11609 | −9 | 11608 | −8 | 11611 | −11 | 12800 | −1200 |

Syene | 13h1/2 | 16800 | 16797 | 3 | 16797 | 3 | 16800 | 0 | 16764 | 36 | 17569 | −769 |

LowerEgypt | 14h | 21400 | 21399 | 1 | 21394 | 6 | 21408 | −8 | 21338 | 62 | 21800 | −400 |

Phoenicia | 14h1/4 | 23400 | 23469 | −69 | 23462 | −62 | 23450 | −50 | 23398 | 2 | 23726 | −326 |

Rhodos | 141/2 | 25400 | 25388 | 12 | 25380 | 20 | 25375 | 25 | 25309 | 91 | 25531 | −131 |

Hellespont | 15h | 28800 | 28798 | 2 | 28788 | 12 | 28817 | −17 | 28711 | 89 | 28800 | 0 |

Massalia | 15h1/4 | 30300 | 30305 | −5 | 30295 | 5 | 30275 | 25 | 30216 | 84 | 30273 | 27 |

Pontus | 15h1/2 | 31700 | 31692 | 8 | 31683 | 17 | 31675 | 25 | 31604 | 96 | 31644 | 56 |

Borysthenes | 16h | 34100 | 34144 | −44 | 34135 | −35 | 34125 | −25 | 34057 | 43 | 34100 | 0 |

Tanais | 17h | 38000 | 37981 | 19 | 37974 | 26 | 37975 | 25 | 37903 | 97 | 38000 | 0 |

S.LittleBritain | 18h | 40800 | 40752 | 48 | 40746 | 54 | 40775 | 25 | 40685 | 115 | 40800 | 0 |

N.LittleBritain | 19h | 42800 | 42761 | 39 | 42758 | 42 | 42758 | 42 | 42705 | 95 | 42800 | 0 |

Residual-Square Sums S(stades squared): | 11279 | 12084 | 8495 | 73470 | 4284816 | |||||||

Residual-Square Sums S(arcmin squared): | 82.86 | 88.78 | 62.24 | 539.8 | 31480 | |||||||

Isotropic Normalized Deviation r: | 0 | 0.89 | 7.8 | |||||||||

Probability P: | 1 | 0.674 | 10^{−13} |

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.,

[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

(Choosing 5' for rounding-precision, might be regarded as a 2nd unknown for the

At the other extreme, the grotesquely outsized

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

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

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

NB: The Neugebauer-Princetitute cubic-polynomial theory was promoted for over 2/3 of a century as the highest wisdom on the matter

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:

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

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
*S*_{m} = 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*,

where K is a constant of little effect in this context.

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** [2009]
‡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.)]

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

In a 1997 addendum, DR commented
(** DIO 4.2** [1994]
[p.57]): “Muffia peddlers of funny
there's-a-Babylonian-in-the-woodpile
explanations for pre-Ptolemy Greek astronomy keep non-citing
(i.e., faking the nonexistence of)
our stark Table 1” [

That there might be occasional individual scholars who fall short here is not the objection. The scandal is that

For 3/4 of a century. And counting.

By not bailing out when contra-Muffia evidence loomed (note common-sense at

Back to reality:

Analysis of our table's data finds *S* is minimized at
ε = 23°37'.60
(primes signify arcmin = 60ths of degrees)
& *A* = −2'.44 (−28 stades,
at Eratosthenes-Hipparchos-** Almajest** scale
of 1° = 700 stades); minimum

As noted above,
Jones (** JHA 33**:15-19 [2002])
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

(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

We now will test Jones'

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" &

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

[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

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,

This dramatizes how spectacularly precise is the fit effected (to a problem of extraordinary sensitivity), by

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 its total integral out to infinity equals unity.)
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* = *x*_{m} & *y* = *y*_{m},

We will call the standard deviation of a single datum σ, which is:

In the case of the ancient klimata, the above 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°51'20" & *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:
*E*^{2} − 3.3126*E* + 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*:

as previously.

Returning to the ** JHA** values and σs:
for the primed-frame normdevs,
the post-rotation equation for
un-normalized

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:

much 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).

Applying it to the ** JHA** proposal is easy
since its

as before.

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

plus the equation for σ
and that for *P* as a function of *r*, we have:

— the very same compact shortcut earlier proposed here.

In passing, we may toss in
a general asymptotic rule, useful when *r*
is large: for any number *U* of unknowns,

where K is a constant of little effect in this context.

Question: Why do we find *U* − 2 in the exponent,
regardless of *U*'s magnitude?

Answer: Because shells in hyperspace have area proportional
to *U* − 1; and integrating, for a range of such shells
at remote *r* values, will obviously knock this down another notch,
producing an asymptotic expression whose exponent contains *U* − 2.

A 3-dimensional isotropic example familiar to physicists is
the Maxwell Distribution for gas molecules' speeds *v*,
where *U* − 2 = 1, as we see from its asymptotic *P*:

with dimensionless *r* related to *v* by the equation

where (in units consistent with *v*'s) *k* is Boltzmann's constant;
*T*, the Kelvin temperature; & *m*, the molecular weight.

The probability density *pd* for the same case
can be precisely expressed as

which isn't analytically integrable in general, though of course the definite integral from 0 to ∞ is unity.

Retrospective Comment on the Foregoing:

The reason our various bivariate 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.

Original Version Uploaded 2009/9/8.

[Occasionally augmented and-or re-edited 2009/9/23-2014/12/29.]

Linked 2009/11/3.