Completion At Last of Centuries of Scholars' Critical Establishment of the Text of the PreEminent Ptolemy


[Geographical Directory]

Afterword by Dennis Rawlins


Diller's Achievement
Aubrey Diller: A Personal Remembrance & Acknowledgement

                                             DR INVESTIGATIONS:
Why a Network of Exactly 360 Sites' Geographical Hours?
The Greek GD's Best Latitudes: Non-Greek Ancient Egypt & Phoenicia
The Unresolved Mystery of Marinos the Phoenician
Astrologers' Handiest Tables, GD Inter-relations, & Accuracy's Degradation
GD 8's Disconnect: the GD a Hybrid
Blest Isles Ignored & Identified: the Cape Verde Islands
Hours as the Route of All Evil in Ptolemy's GD
Precession and Aristarchos
Marinos Mis-Dated?
Tyre: the Missing Home-City of Book 8's Supposed Source
Landlubber Ho! Ancient Saigon & Hanoi; Wrapped China Negates Pacific
Brief Comments & Hypotheses on Several Subjects
1st World-Map Projection: Wherefrom the 34? Ancient Averaging. Weights?
The Impossible Dream of the Symmetric-Rectangle-Bounded Ekumene Fan

Diller's Achievement

  • Aubrey Diller (1903-1985) was long the world's leading philologist in the area of ancient geography. Which is one of several reasons why our 2006/12/15 publication here of DIO 5 [2006], Diller's edition of the final book (Book 8) of the legendary 2nd century AD Geographical Directory, marks something of a milestone:

  • All other important parts of Ptolemy's Geographical Directory (henceforth GD) have previously been published in critical editions: see list, below.
    [As long ago as 1929, the eminent classicist Ernst Honigmann was publicly calling for accomplishment of a reliable edition of Book 8.]

  • Thus, DIO's publication of Diller's rendition of Book 8 ensures that for the 1st time since the end of classical antiquity, scholars finally have access to the complete, critically-established text of the full GD — from the start of Book 1 Chapter 1 right through to the end of Book 8.
    [At the time of this posting the wonderful Stückelberger-Graßhoff-Mittenhuber edition of the complete GD also appeared (Univ.Bern 2006), a beautiful and thorough labor of dedication.]

  • It is astonishing that it has taken 2 centuries for the text-establishment process (begun c.1800) to be completed — after all, the GD is THE most written-about geographical work in the history of the field. (See, e.g., Wm.Stahl Ptolemy's Geography: a Select Bibliography NY Public Library 1953.)

  • A partial list (emphasizing scholarly editions) of the other GD parts' publications:

    J. Lennart Berggren and Alexander Jones (GD 1 & 2.1) Ptolemy's Geography Princeton 2000 pp.57-95, English.
    Carl Müller (GD 1-5) Paris 1883&1901, Greek & Latin.
    (Thorough [actually overthorough] if non-ideal apparatus.)
    Friedrich Wilberg & Carl Grashof (GD 1-6) Essen 1838-1845, Greek & Latin.
    (Scrupulous apparatus.)
    Karl Nobbe (complete GD 1-8) Leipzig 1843-1845, Greek. (Very brief [purely verbal] apparatus at 3:190-207. But complete [all 8 books] and exhaustively indexed for both geographical sites and Greek words.)
    Louis Renou (GD 7.1-4) Paris 1925, Greek & French.
    (Thorough apparatus.)
    Otto Neugebauer (GD 7.5) Hist Ancient Math Astron Springer 1975 (data discussed p.935), English. (Not a translation of the chapter.)
    Otto Neugebauer (GD 7.6-7) Isis 50:22 (1959), English.
    J. Lennart Berggren & Alexander Jones (GD 7.5-8.2) Ptolemy's Geography Princeton 2000 pp.108-121, English.
    Alfred Stückelberger, Gerd Graßhoff, Florian Mittenhuber, Renate Burri, Klaus Geus, Gerhard Winkler, Susanne Ziegler, Judith Hindermann, Lutz Koch, Kurt Keller (complete GD 1-8) Handbuch der Geographie, Bern University's Institute for Classical Philology 2006, German.

    Aubrey Diller: A Personal Remembrance & Acknowledgement

  • I first conversed with Aubrey Diller by telephone from San Diego on 1979/11/26. (At the time, he was Prof. Emeritus of the Indiana University Classics Department.) I was delighted to impart to him the startling news of independent vindication of his 1934 solution to Strabo's Hipparchan klimata (which had then-recently yet-again been [unjustly] denigrated, in Otto Neugebauer's [justly] valued History of Ancient Mathematical Astronomy [HAMA] Springer 1975 p.734 n.14) — proving that these klimata were computed by spherical trigonometry (just as Ptolemy later computed them). For the original 1934 paper, see Aubrey Diller. Klio 27: 258-269: “Geographical Latitudes in Eratosthenes, Hipparchus, & Posidonius”.

  • There are now various proofs that Diller was right. See D.Rawlins Publications of the Astronomical Society of the Pacific 94:359-373 [1982a] p.368. Also R.Nadal and J.Brunet 1984. Archive for History of Exact Sciences 29:201-236 [1984]. And DIO 4.2 [1994] p.56 Table 1. Detailed discussion & update, plus repeated unanticipated discovery of new data that perfectly fit-confirm Diller's theory, at: and thereabouts.

  • Diller later reported (1980/1/24 letter to DR) his happy astonishment at the crucial 1934 discovery's surprise-reincarnation:

    It was somewhat dramatic for my old article, after 45 years of oblivion or abuse, to be rescued by a phone call from a stranger in San Diego. I thank you for the recognition.

  • We were never strangers after that.

  • His letter went on to say that his approach was primarly non-mathematical. (Considering the import of his 1934 solution, I always thought Diller much too modest about his math ability.) “I spent a good deal of time on the manuscripts in Europe in 1950-2.”

  • Diller kindly sent (1980/2/11) a copy of the G.Olms 1966 reprint (Hildesheim, Germany) of the old Karl Nobbe 1843-1845 edition of the GD (then still the sole complete respectable edition, whatever its shortcomings), for which Diller had written the (Latin) Preface.

  • A few years later, I moved back to my native Baltimore. During the long 5000 km cross-country drive, I stopped to see only one person: Aubrey Diller. On 1982/8/4, I phoned him from Illinois to say that I could drop by for a visit that evening — and (after a dangerous highspeed night-drive through a severe rainstorm) I finally arrived at his apartment at 11:30 PM, dressed in a huge cowboy hat [relic of crossing the Rockies with minimal sunburn] and shorts! The door opened, and Diller appeared: such a civilized gentleman, that I am amazed he welcomed the apparition I indubitably presented to his sensitive sight. He was a frail-looking scholar with very long hair — but I soon realized that, though fastidious, he was anything but frail when expressing his opinions. (During comments on modern morality, he asked if I'd ever been divorced and seemed relieved that I hadn't: “It's so cheap.” Emphasis in orig.) We had such a glorious good time talking — mostly about our mutual fascination, ancient geography — that it was 2:30 AM before we parted.

  • But the issue of this fateful meeting was to be more than a lifelong friendship. As we chatted on 1982/8/4-5, I learned that Diller had for years been quietly compiling a vast index-card file of all mss' data for Book 8 of the Ptolemy Geographical Directory. I knew that the material was vital for evaluating the basis of what survives of ancient mathematical geography.

  • Soon after, I was able to put Diller's Book 8 gatherings to good use in a talk at the Longitude Zero Symposium (held at the National Maritime Museum in Greenwich [London, 1984], commemorating the centenary of the Greenwich Meridian — DR's paper reprinted at Vistas in Astronomy 28:255-268 [1985]), describing his theory that Book 8 or equivalent was the ultimate math-foundation of the grid-network underlying Books 2-7. (The germ of the theory was initially published in D.Rawlins Publications of the Astronomical Society of the Pacific 94:359-373 [1982a] p.368.)

  • It should be noted that Diller himself never accepted this DR theory — as I learned during a chat of 1983/1/26 — though he was far from adamant on the issue.

  • Diller also knew of my views on Ptolemy's integrity, but he had no strong opinion on that controversy, and it never came between us in the slightest.
    [Note contrast to certain religious & cultist fanatics.]

  • At our first meeting and later, I urged Diller to put together a completed edition of Book 8: preface & data. Despite a 1983 heart attack, he warmed to the idea, and I liked to think that (in his 80th year), such a project as we contemplated would be healthful as well as fruitful.

  • Diller found that he needed one more original GD ms, the Seraglio manuscript (Istanbul) which the Newberry Library (University of Chicago) fortunately possessed photographs of. With the kind assistance of the Newberry's Lucille Wehner and David Buisseret, I arranged (and paid) for a microfilm to be sent Diller (1983/8/18 & 9/21).

  • I visited Bloomington 1983/10/30-11/3, and we spent alot of time with each other. Diller even drove me to see his favorite bucolic scenes. (He was quite a walker.) And we took a day to visit the wellknown Brown County Art Gallery.

  • We usually ate together in the IU dining room. I learned that he had stopped smoking many years ago because it caused headaches. We agreed that without those headaches, he'd probably be dead long since.

  • Diller was a lifelong bachelor whose manner probably seemed abrasive to some. He was never reticent about his vast amusement at my amateurish mangling of Greek pronunciations. (Though I feeble-defensively asked how we know what ancient pronunciation was like.) But such interaction never got in the way of good fellowship. Though not a professional mathematician, Diller was far better at math than I will ever be at any foreign language. Regardless, between us, we had (in extreme shares) all the talents requisite to doing a full edition of the Ptolemy GD, and it is unfortunate that Diller could not live long enough to join me in such a project.

  • I happened to phone Diller on 1984/9/3, when he had just that very day essentially completed the edition of Book 8 that appears here. He sent it to me on 1984/10/3 — which was a valued token of trust, in a day when plagiarism is not as uncommon as it ought to be.

  • Aubrey Diller passed away the following year. I am grateful and honored that Diller left his ultimate scholarly contribution to DIO. Its publication here marks the fulfillment at last of a promise made to him years ago. I hope that the result will add to the lasting memory of one of the purest scholars of his & my century (the 20th), whose friendship was to me at once a pleasure and a privilege.

    Notes added 2006/10/15:

  • The Diller-GD project has now lain much too-long unpublished, but this may prove fortuitous in that it has allowed the young yet already eminent classicist (and rare scholarly-gem discoverer) Alexander Jones (University of Toronto), to vet the release of the work. It has also allowed fermentation of DR's theories of the GD's evolution. See occasional observations below.

  • A few minor clarifiers:
    [a] The word ekumene (oικoυμενης) — from which the ecclesiastically-ambitious word “ecumenical” derives — should be translated: the inhabited or known world.
    [b] Also, DR would of course use “latitude” at some of the places (in GD 8.1-2) where Diller's translation says “declination”.
    On such points, the Berggren & Jones 2000 pp.118-121 translation of the brief G8 preface is superior to Diller's. (Though, keep in mind that the Diller ms was to be jointly edited by us [a project cut off by his death], which would have removed astronomical ambiguities.) Regardless, Diller's great GD contribution is his pioneer establishment of the mass of Book 8 numerical data. Only one person can perform such a deed first. This alone would merit his scholarly immortality — which is virtually overkill, since he had already assured his eternal remembrance by his mathematical detection of the scheme underlying the Hipparchos-Strabo klimata, revealing use of sph trig earlier than anyone had known (until Diller's 1934 paper), as well as the best ancients' adoption of an accurate obliquity (23°2/3, correct within c.3') which had not survived in mss that came down to us.
    [So long as the cults of Gingerich (AAS-HAD & JHA) and Neugebauer continue to remain unable — through their group-intellect's failure of sharpness and-or independence — to recognize (even the possibility of value in) this epochal and repeatedly surprise-vindicated-on-the-nose Diller double-discovery, DIO will continue to regard them with the contempt which cultish mind-lock and censorship invite.]

  • Thanks are due here to DIO Editor, Dennis Duke, for getting the GD Book 8 project re-started early in 2006, as well as for restoring the original 1984 Diller ms to a publishable state. Expert advice from Alex Jones and Len Berggren headed off potential mis-steps. Also to be thanked is a longtime family friend, the late Prof. Emeritus Jimmy Poultney (for many years one of the stars of the Classics Dep't of the Johns Hopkins University), who kindly oversaw DR's early work on Diller's final opus; and David Rockel, who patiently assisted in the collection of materials used in DR's research.

    Finally, with a sentimental mix of sadness and gladness, we note that this marks the 4th time that DIO has published the final paper of one of the world's pre-eminent scholars in his areas — the plural denoting that each was, remarkably, a leading figure in more than one field:
    Robert Newton
    David Dicks
    Hugh Thurston
    Aubrey Diller

    of the

    by Dennis Rawlins

    Distillate from 3 Decades of
    DR Researches into the GD

    [In 2008, paper copies of these researches were sent to our library subscribers, in DIO 14 ‡3 [pp.33-58].
    Readers who wish their own booklet paper copy (of this or any DIO) may download the appropriate pdf, gratis.
    [This version (or the corresponding web pdf) includes [p.58] an extensive References section.
    It also includes [p.50] a postscript rendition of the GD's projection of the inhabited world (ekumene), which fulfils the originator's vision more perfectly than the Nobbe version (reproduced here) or any other previously published (since antiquity, at least).]

    Unless otherwise indicated, GD section-numbering here follows that of the Nobbe edition (Leipzig 1843-1845), numbering which has fortunately also been adopted as closely as possible by the excellent new Stückelberger & Graßhoff 2006 edition.
    Note that the present paper forgoes the use of accents for Greek words here. Diller himself pointed out their superfluity, since classical-era Greek lacked accents. During a DR 1987/6/1 visit to the Vienna Papyrus collection, the same view was expressed by the head of the collection, as well as by the able Dutch scholar Peter Sijpesteijn, who happened to be visiting the same day.

    Why a Network of Exactly 360 Sites' Geographical-Hours?

  • The famous Ptolemy Geographical Directory (henceforth GD), popularly called “The Geography” (sometimes “Geographia”), is in Eight Books. The GD begins with explanatory Book 1, which tells of Ptolemy's incorporation of thousands of sites' geographical places from the work of an earlier geographer, Marinos of Tyre. Then, Books 2-7 list about 8000 sites' positions, expressed consistently in degrees: longitudes in degrees east of the Blest Isles (Cape Verde Islands), and latitudes in degrees north or south of the Equator.
    [DIO's people are amazed at a long tradition of suggestions that the GD may well be the earliest geographical work ever to use spherical coordinates. This is less scholarship than a relic of Neugebauer-salesmanship for Ptolemy. (Origin: O.Neugebauer, HAMA 1975 pp.337, 846, & 934; and see p.280 for parallel celestial semi-claims for the Almajest, despite Hipparchos' 2nd century BC Commentary's listing of dozens of stellar Right Ascensions & Declinations.)
    [Long before Ptolemy, Strabo reported a Nile map consistent with use of spherical geographical coordinates, and which goes back at least to Eratosthenes (3rd century BC) — a map so antique that it does not even use degrees.]

    The GD then concludes with what DR contends resembles and-or partly constitutes the data-base grid-network underlying the precision-corrupted positions of GD Books 2-7, namely: Book 8, whose data are expressed entirely in hours (not degrees), a list of 360 sites' longitudes in hours west or east of Alexandria and (instead of latitude) longest-days (for Summer Solstice) in hours, where parallels at 1/4-day intervals of M were called “klimata”. E.g., longest-day M = 14h1/2 was called the Rhodos klima where L = 36° (via the below equation).

  • Diller was (1983/3/6 letter to DR) the 1st scholar to point out the 360-site total and to suggest its deliberateness.
    [The total of his lists is 359. Nobbe's total is 358. But Nobbe omits Tarentum and Sousaleos, while Diller semi-omits Limyra. Merging the lists, we have exactly 360 sites in 26 sections, corresponding to GD 2-7's 26 maps. Sections: 10 of Europe (118 GD 8 sites), 4 of Africa (52 GD 8 sites), 12 of Asia (190 GD 8 sites).
    I propose scholars' agreement upon a conventional numbering of all 360, based upon the sequence of Diller's XZ Codices, dovetailing with the UNK Codices (to cover sites either skipped), which follows Diller's desire to give primacy to the former. We use prefix D, to number every GD 8 site, so that “D x” refers to the xth site. Adding to Nobbe's edition of GD 8: Tarentum (GD 3.1.12, 8.8.4) as site D53, Sousaleos (GD 3.3.4, 8.9.3) as site D63. (Note that we are dovetailing these two sites into Nobbe in passages that [exceptionally] already list more than one site — which may help explain these two oversights.) To Diller's version, we add Limyra (GD 5.3.6, 8.17.25) as site D193, Diller XZ Codices Asia-Map 1 site #22→#22a: “Myra”, whose coordinates are identical to Nobbe's “Limyra” at GD 8.17.25. D192 is UNK's item#22, whose coordinates are identical to Nobbe's GD 8.17.23, “Myra” (GD 5.3.6). Note that one finds “22a” in Diller's hand in the left margin of his p.X13, showing that he suspected the need to add this site as the final touch to perfecting his epochal document. I.e., even at age eighty-plus, his sharp eye was still missing nothing!]

  • The longest-day M (in hours) at a site is a sph trig function of latitude L (in degrees) and the Earth's obliquity ε (also in degrees), by an equation known at least since the 2nd century BC (Bithynians Theodosios & Hipparchos) — a remarkable historical revelation, which we recall is primarily owed to Aubrey Diller's 1934 mathematical investigation.
    [Readers not into sph trig may now skip the next few paragraphs.]
    Th equation for computing each klima is attested for the 2nd century AD at Almajest 2.3:

    cos(15M/2) = − tanL tanε

    (where obliquity ε was usually taken to be 23°5/6 or [Diller's discovery] 23°2/3).

  • Why different data-format for GD 2-7 vs GD 8? Two possible answers:
    [1] Books 2-7, like the important-cities part of Ptolemy's Handy Tables, are in the form of Marinos' manual or map, presumably after his (?) systematic tectonic mass-alterations (GD 1.4 & Berggren & Jones p.46) to force macro-geographical accord with the above-hypothesized network-grid-basis, which had been severely pre-corrupted by astrology tables' roundings.
    Remarks at, e.g., GD 1.18 suggest that, like (following?) Hipparchos, Marinos clumped cities under parallels. Also, Marinos gave primacy (ibid 1.20) to Hipparchos' 36° parallel through the east-Mediterranean island of Rhodos, suggesting both an astrological tradition and even the possibility that his table of parallels (for at least the Mediterranean-region) was a hand-me-down from Hipparchos, whose main observatory was located on Rhodos (D149), probably just north of the town of Lindos. (See DIO 4.1 [1994] ‡3 §F [pp.42-45].)
    [2] The data of Book 8 are not for a map — but are in precisely the hour-based form for astrologers' convenient use in computing a horoscope for a site other than Alexandria (D149), which was obviously the standard meridian for astronomical & astrological ephemerides in the Hellenistic world.
    [The very choice of longest-day (instead of latitude) as GD 8's measure of northerliness tips us off to the astrological connexion. (Hardly a stretch: recall that Ptolemy compiled the superstitious horoscope-delineation book that is still astrologers' bible: the Tetrabiblos. Note that the geographical table in his astrologer-oriented Handy Tables was at this stage still inconveniently in degrees. So GD 8 could have been called the Handiest Tables — perfectly set up for astrologers' convenience. Listing cities by longest-day superficially appears odd & cumbersome, and it gave no special aid when using data for maps. To the contrary. However, astrological tables of the outdoor-invisible “Ascendant” (Almajest 2.8) — the corner-stone of astrological “house”-division for horoscopes — were simply easier to computationally construct in the 1st place (math provided at ibid 2.7), for longest-day values than for latitude values, back in the ultra-longhand days of early use of sph trig.
    The computational utility of longest-day is easy to show.
    For A = Ascendant & ST = Sidereal Time, one can calculate:

    cotA = (cos[15M/2] − sinST)cosε/cosST

    which is simpler than using latitude L with a formula based upon combining the foregoing two equations:

    cotA = − (tanL sinε + sinST cosε)/cosST

    (Astrologers' other key invisible celestial point was the “Midheaven”. But the Midheaven MH is latitude-independent. So, for any latitude, one need only consult [in Alm 2.8's tables] the “Sphaera Recta” columns [Toomer 1984 p.100], which tabulate the relation tanMH = tanST/cosε.)
    Thus, most available ancient astrological tables of houses (e.g., Almajest 2.8) were arranged by klimata, i.e., longest-day. It will help to provide an example, using Almajest 2.8 for Rhodos (D189) at Sidereal Time (the Right Ascension of the meridian, or Hour Angle of the Vernal Equinox) 21h23m36s = 320°54' (which is chosen to avoid interpolation in step 1, as will be evident):
    Adding 6h or 90° gives 50°54' (the rising point on the Equator). Then, find 50°54' in the Alm 2.8's “Accumulated Time-Degrees” column for Rhodos (longest-day M = 14h1/2, the basis of this column's ancient computation and arrangement): Almajest  2.8 (Toomer ed. 1984 p.101). The value on the same row in the column “10° Intervals” is zodiacally 10° of Gemini or 10°GEM 00' = ecliptically 70°, so that is the Ascendant. The Descendant (ecliptical point that is setting) is opposite: 250° or 10°SGR 00' (10° of Sagittarius). The Midheaven (polar longitude of transitting zodiac point) is then found by linear interpolation on Toomer 1984 p.100: in the “Accumulated Time-Degrees” column, under the “Sphaera Recta” heading, we find 312°32'; 320°54' (ST) exceeds this by 8°22' of the 9°58' interval corresponding to the 10° interval between 10°AQR 00' and 20°AQR 00' (in the column “10° Intervals”), so: add 10°(8°22'/9°58') = 8°24' to 10°AQR 00', which yields Midheaven = 18°AQR 24' (18°.4 of Aquarius) on the zodiac or ecliptical longitude 318°24'. The Nadir is opposite: 138°24' or 18°LEO 24'. (This establishes the four cardinal points of the astrological houses for the chosen place & time. Division of each quarter into three parts then establishes the twelve astrological houses, but said division differed from one house system to another. Sph trig-based tables of houses probably go back to Theodosios of Bithynia, 2nd century BC.) Note in passing: finding Ascendant & Descendant (and thus house-divisions) is the sole use most modern astrologers have for geographical latitude. (Ancients also used latitude to enter parallax tables, but such scrupulousness is rare among today's astrologers.) Geographical longitude was used merely for additively converting local time to ephemerides' standard zero-meridian, presumably that of Alexandria.]

    The Greek GD's Best Latitudes: Non-Greek Ancient Egypt & Phoenicia

  • Fortunately, some cities' accurate latitudes appear to have survived; two particular groups are consistent (if we include refraction and rounding) with that optimistic conjecture.
    In Lower Egypt (DR 1985 p.260; GD 4.5.53-55): Memphis (D151), Cairo [fortress “Babylon”], and On [Heliopolis]. Note (in the context of astronomy-based latitudes): these 3 sites cluster around the most accurate astronomical-surveying-oriented building of antiquity, Giza's Great Pyramid, whose latitude is correct at GD 4.5.54 (Cairo [Babylon], thus adjacent Giza): 30°00'N.
    [All three latitudes are correct, perhaps a notable Egyptian achievment — since the GD lists Heliopolis (the Greek name for On) at the wrong latitude (exhibiting a peculiarly Greek error), not realizing that it is the same place as the holy city called “On” by the Egyptians and (Gen.41.45) the Bible. Note that the correct latitude is associated with the ancient Egyptian name, not the later Greek one. Details at DR 1985 p.260.]
    In Phoenicia (modern Lebanon): Acre (Ptolemais), Tyre, & Sidon have errors of only a few miles, not quite as right-On as the Egyptian trio, but nonetheless impressive for antiquity — and highly unusual for the GD, suggesting that Marinos in Phoenicia (like Hipparchos at Rhodos) got particularly accurate latitudes from his own observations or from those of local astronomers or navigators, even while absorbing and relaying ordmag 1° errors for regions outside his or his associates' direct experience.
    Of these six sites, only Memphis (D151) is listed in GD 8.
    [Memphis' XZ (ms-tradition) longest-day (14h) appears independent; but the ultra-precise UNK value (13h57m) looks like it may have been adjusted-to (computed-from) an accurate latitude — suggesting post-Ptolemy tampering. See the learned observations of Berggren & Jones 2000 (p.44) upon the two ms-traditions' relative trustworthiness and purity.]

  • The implication: those major cities (civilized enough to desire and afford astronomers) which are not listed in GD 8 show a better chance of having accurate GD 2-7 latitudes than those which don't.

    The Unresolved Mystery of Marinos the Phoenician

  • Why hasn't it been previously noted that GD Book 1's extensive critical discussion of Mediterranean-region scholar Marinos' data fails to provide unambiguously a single Marinos latitude in degrees for any Mediterranean city? — or, indeed, any city within the Roman Empire.
    [Could Ptolemy have made the omission deliberately? I have doubts on that point; however, such silence would be similar to the slyness evident in his Almajest 3.1 suppression-silence regarding the times of the solstice-observations of Aristarchos (truncated) & Hipparchos (good to 1h!), omissions 1st stressed by the late W.Hartner (letter to DR). See DIO 1.1 [1991] ‡6 §§A5&B3-5 [pp.50-52]. Note the key correlation: these two solstices are the only members of Ptolemy's extensive set of times of solstices & equinoxes that do not agree with his (Hipparchan) tables, and they are the only ones for which he hides the hour. (Each disagreed with his tables by 1/4 day.)]

  • So, though Marinos' latitude for Okelis (D281) hints at inferior accuracy, we cannot tell for sure whether his important-city latitudes were as corrupt as the GD's. I.e., Ptolemy's silence on Marinos' latitudes within the Roman Empire leaves open the possibility that Marinos' latitudes for Mediterranean or Roman Empire sites were accurate (if so, GD data-degradation occurred after his time) — and were thus suppressed for disagreeing with those of Hipparchos. But would encyclopedist Ptolemy expend the huge effort required for shifting 8000 data to dovetail with an underlying the grid-network's few hundred important cities? (Ptolemy does report [GD 1.18] that much of Marinos' data for minor cities were incomplete and-or scattered, so serious labor [on someone's part] was required for subsequent estimation of positions' precise longitude & latitude.) Yet, if Marinos were an astrologer, why would he give longitudes in degrees — and worse, in degrees from the Blest Isles, not Alexandria? (Yet, Ptolemy's astrologer-fave Handy Tables did likewise.)
    The actual purpose of using the Blest Isles as longitude zero was probably to eliminate east-west positional sign-ambiguities — just as NPD does for north-south.]
    With arguments available in both directions, it is hard to be sure how much responsibility (for the corruption of GD 2-7's latitudes) is borne by Marinos.
    In favor of Marinos being a geographer, not an astrologer, is the measure of longitude in degrees from the Blest Isles. Which in turn implies that key sites' latitude-corruption from rounded longest-day klimata was not Marinos' doing.

  • After all, how could it be that an (apparently) eminent geographer from Phoenicia (a legendary naval center, for whom latitudes and stellar declinations would have been vital for navigating commercial vessels if nothing else) was ultimately — via his own or others' sph trig — depending, for his latitudes, upon crudely-rounded astrological tables? (Of longest-day data: see below.) If he was. Note that the Marinos-of-Tyre-based GD 5.15.5&27 latitude of Tyre is just about exactly correct (to its 5' precision) if based upon observations of circumpolar stars (affected by c.2' of atmospheric refraction), a wise and effectively parallax-free latitude-determination method which may go back to the time of the Great Pyramid. (See Rawlins 1985 pp.255-256, as well as D.Rawlins & K.Pickering Nature 412:699 [2001/8/16]; see also DIO 13.1 [2003] [pp.2-11.)
    [Similarly, Hipparchos knew his own latitude, but seems to have been weak elsewhere, e.g., placing Athens a degree south of its actual latitude (Commentary 1.11.3&8) and Babylon 2°1/2 north of its — both values copied by the GD.]
    So: what was the purpose of Marinos' geography? Naval? Or natal?
    If latitudes based upon longest-day data were Marinos', this would raise the suspicion that he was an astrologer. (Possible, but — as already noted — his reckoning longitude in degrees and from Blest Isles is contra this idea.) Were famous ancient astrologers analogous to modern popular-science writers and publications, where ubiquity, lucre, and hype obscure innumeracy, thereby nourishing blind-leading-blind multi-generational replication of unreliable scholarship?]

    Astrologers' Handiest Tables, GD Inter-relations, & Accuracy's Degradation

  • GD 8.2.1 states that the data of Book 8 were computed from latitudes & longitudes.

  • However, a detailed mathematical case has been made by D.Rawlins Vistas in Astronomy 28:255-268 [1985] that (though the remote-past origin of longest-day M data were obviously computed from latitudes) the highly corrupted latitudes of major cities listed in GD Books 2-7 must have been computed (via the 1st of the above equations) from conventionally over-rounded M-data of just the sort we see in Book 8.
    Flagrant examples appear below, e.g., for Babylon and Vietnam. [This is not to deny that some GD calculations went in the other direction — nor even to reject the distinct possibility that GD 8 was all computed from GD 2-7 (data themselves already corrupted by calculations from a prior pool of longest-day data) as alleged. But some differences in the two mss-traditions (Diller's XZ vs UNK) occasionally remind us that post-2nd-century AD revisions of the GD 8 values may have attempted arranging consistency, in the same spirit that latitudes in GD 2-7 were adjusted at some point (at or before Hipparchos' era), according to the DR theory of the GD.]
    The suggestion here is that distortions in GD latitudes go back at least to Hipparchos. (Note that this theory has here been limited to proposing the high likelihood that data of the sort provided in GD 8 underlay GD 2-7's major cities.) The distortions in longitude probably occurred later than Hipparchos, since they involve a shift from the Hipparchos 252000-stade Earth-circumference to the 180000-stade Earth-size which fellow-Rhodian Poseidonios seems to have switched to (Strabo 2.2.2) during the 1st century BC. (Though C.Taisbak eruditely wonders if this switch wasn't much later.)
    The Almajest was still using the larger Earth-size during the mid-2nd century AD, and the earliest rock-certain attestation of the smaller value's use is by Marinos, around the same time.
    [Berggren & Jones 2000 p.14 n.10 show excellent judgement in rejecting a misguided but persistent tradition of manipulating the stade, to force disparate ancient Earth-measures to agree with each other or reality. See also Rawlins Archive for History of Exact Sciences 26 pp.211f (1982); DIO 6 [1996] ‡1 §C14 & n.47 [p.11].
    The unpopular but evidentially-insistent fact that Eratosthenes' Earth-circumference was genuinely (not illusorily) high by 1/5, and Marinos-Ptolemy's too low by 1/6, is shown by 3 considerations:
    [1] Ptolemy's expansion (by over 1/3) of the longitudinal distance from Rome to Babylon between Almajest  and GD.
    [2] The GD's similarly large (33%-40%) systematic over-estimate of actual longitudes. (See DR's least-squares test at Vistas 1985 p.264, leading to p.265's table of reconstructions.)
    [The first scholar to sense that ancients had multiplied longitudes by adjustment-constants (when adopting new Earth-sizes) seems to have been Pascal Gossellin Géographie des Grecs 1790. (See his several tables exploring this hypothesis; also DR 1985 n.22, which credits Gosselin & van der Waerden for this important realization.)]
    [3] DR's neat common explanation of both C-values' errors from atmospheric refraction of light with 1/6 the curvature of the Earth.
    All 3 of these evidences are consistent with each other and with realization that Ptolemy (or his source[s]) adopted the genuinely smaller Earth entailed by his 700→500 stades-per-degree shift. Thus, there is not only no case-for but no longer even any need-for the literature's ever-reappearing attempts to claim that Eratosthenes got-the-right-answer for the Earth's circumference but expressed it using an undersized stade.]

    (Columbus' belief, that the shortest trip to China's Kattigara [D356] was westward not eastward, was much influenced by Marinos' over-tiny Earth.) Conversely, there are plenty of hints that the majority of GD 8's non-major cities may have been directly computed from data of the sort found in GD 2-7. (Note strong evidence that neither section was directly computed from the other.) E.g., the greater precision of GD 2-7 data is obviously often impossible to derive by computing from GD 8, while the reverse is seldom impossible. Further, late copies of Ptolemy's Handy Tables (a work probably earlier than the GD) contain a list of c.360 Important Cities' (364 in Halma's ed.) latitudes and longitudes in degrees, very similar (though not identical) in content, bulk, and sequence to GD 8. It may be that Ptolemy simply computed the non-key sites of GD 8 from something like this list, as a handiest-possible add-on to crown his GD.

  • However GD 8 was accomplished, it was an astrologer's-dream Handiest Tables,
    the only example of its type that survived from classical antiquity:
    [1] All latitudes expressed in longest-day, for easy entry into tables of houses.
    [2] All longitudes expressed vis-à-vis Alexandria, and
    [3] in hours — for converting local time to Alexandria time, to enter Alexandria-based tables for computation of the zodiacal positions of Sun, Moon, & planets.

  • Berggren & Jones 2000 p.29 notes (as did Rawlins 1985 pp.261f) specific cases where key cities' latitudes must have been computed from longest-day.
    [A semi-ambiguous detail: Almajest 2.13 refers to the upcoming GD and refers to degrees vs the Equator for latitudes (like GD 2-7), but it speaks of placing sites by degrees (the measure of Books 2-7), while using the Alexandria (D149) meridian of Book 8 and of E.Mediterranean astronomers & astrologers; so it conclusively favors neither side on the relation between the GD's two data-sections.]
    Regarding the preface to GD 8, it should be added that:
    [a] The preface's comments on map-distortions belong among the mass of such material set forth back in GD 1.
    [b] One of the most obvious arguments against GD 8's data being for (non-warped) maps is that longest-day data are not linearly related to latitude. (Note shrinking of klimata-bands with recession from the Equator at, e.g., Stückelberger & Graßhoff 2006 2:748-751.)
    [c] The GD's regional maps have come down to us. Granted, they are not originals; nonetheless, their fidelity to the GD's regional dividers strongly suggest that these are the originals in essentials. Though the maps' margins bear longest-days marks (inevitably at large latitude intervals), the densely-marked, dominant north-south co-ordinate (linearly related to up-down distance on each map) is latitude in degrees. Which is necessary because these maps depict the locations of thousands of cities (not the hundreds of GD 8), the great majority of whose positions are not given at all in GD 8, while all their longitudes and latitudes are in GD 2-7. More indicative yet, the maps measure longitude not in GD 8's hours east or west of Alexandria, but in GD 2-7's degrees east of the Blest Isles.
    (See the beautiful reproductions of several such maps between pp.128&129 of Berggren & Jones 2000.)
    So: why would GD 8's preface be discussing the construction of regional maps actually based upon the data of GD 2-7?
    Is this more residual evidence of patch-work authorship?
    Question-in-passing: What evidence is there that Marinos had anything to do with constructing GD 8? The absence of his native coastal Phoenicia from GD 8 proves his non-authorship of it.
    [In Nobbe's edition, at GD 8.20.18 (Jerusalem D247) the spelling of “east” changes from ανατoλας [anatolas] to εω [eo] for most of the rest of GD 8. If this is meaningful (and mss disagree), it is possible that this is connected [a] to the compiler's departure (at about this point) from a map of the Roman Empire to an extra-empire map of different format (and less reliability), and this perhaps led [b] to the accidental omission of coastal Phoenicia, possibly due to the two maps' different order of site-listing around the nearby seam.
    More patch-workery?]

  • Tyre's absence from GD 8 only adds to the evidence that GD 8 is not directly connected to Marinos-of-Tyre's Books 2-7. So it would be wrong to over-claim that GD 8 is the father of GD 2-7. Uncle or cousin might be nearer the mark. For, longest-day data (the stuff of GD 8) are obviously the basis of the full work's flawed grid-network of important-cities' latitudes — a grid which typically misplaced geographically-key cities by ordmag a degree, grossly mislocating the latitudes of, e.g., Byzantion (D87 [Istanbul]) by 2° (though, as Berggren & Jones 2000 p.29 n.37 rightly marvel, the false GD latitude continued to be believed at religiously non-empirical Byzantion until c.1000 AD!); Carthage (D131) by 4°, a huge error (revealed at DR 1985 p.263 as due to false M) that enormously distorted maps of N.Africa (up to the Renaissance, over 1000 years later). Not to mention Babylon (D256) by 2°1/2 (idem n.13) — a discrepancy which is difficult to reconcile with a modern historian-cult's non-empirical insistence that Greece had high-astronomy debts to Babylon.
    [It is common knowledge that the longest-day value (GD 8.20.27) for Babylon (D256), 14h5/12, is a rounding of 14h2/5 — which is 3/5 of a day and the M basis of computation of the revealingly inaccurate latitude L = 35°N (GD 5.20.6).]
    DR suspects that the latitudinal shortcomings of the GD's basic grid-network derive primarily from astrologer Hipparchos (not Marinos or Ptolemy): see hint at GD 1.4.2 (& 8.1.1), where is cited Hipparchos' listing of cities under-the-same-parallels (“klimata”) — obviously for astrologers' convenient entry into common longest-day-based tables of houses. This step typified the fateful Ptolemaic sloppiness which (DR's theory proposes) was the prime source of latitude-accuracy's corruption in GD 2-7.
    [There remains the question of whether Hipparchos was responsible for the fateful step of converting crude tabular longest-day M values from hours to degrees of latitude L. In the light of DR's 2007 realization of just how admirably accurate Hipparchos' longitudes may've been, the odds that he was not the culprit are enhanced.
    Has the remarkable irony been noted that the Geographical Directory (at GD 8.1.1) itself scoffs at the common folly of clumping cities under parallels? Or that this contradicts GD 1.4.2, where Hipparchos is praised for his alleged aloneness in performing the very same clumping? Of course, the GD 8.1.1 complaint is merely that parallel-lists [like the pre-Ptolemy one of Pliny (77 AD): analysed at Rawlins 1985 p.262] waste time and space, but the statement is valuable in its suggestion of ancient currency of the very lists upon which the DR theory is founded.
    (We needn't speculate anyway, on the existence of lists of a few hundred key cities' coordinates. Just such a list survives, e.g., in the Ptolemy Handy Tables, the “Important Cities” table of which (N.Halma 1:109f [1822]), appears closely related to GD 8 in both quantity and sequence: 364 sites in all, with 12 not in GD 8, and 8 missing in HT. See also the two Important Cities lists provided in E.Honigmann's Die Sieben Klimata 1929 [pp.193f]: Vatican 1291 [493 sites] and Leidensis LXXVIII [a comparable number of sites]. These lists' positions are [like GD 2-7] given entirely in degrees east of Blest Isles and north of the Equator.)
    Said currency could help a defense of Hipparchos as not-necessarily the unique source of the GD's macro-errors; however, his fame and his citation by both Marinos (GD 1.7.4) and Ptolemy (Almajest passim) argue in favor of his culpability here, though see above speculation on his longitudes.]

  • Rounding klimata to fractions of hours (Book 8's practice) correlates to FAR cruder precision than rounding latitudes to twelfths of degrees, which is the precision of Books 2-7's data. Ancient longest-day tables often rounded longest-days to the nearest quarter-hour or so. (See, e.g., Almajest 2.6, HAMA 1975 pp.728f.) But when using sph trig in the Mediterranean region, a longest-day error of merely five timemin would cause an error of nearly a full degree — which is the actual (terribly crude) accuracy of the data of Books 2-7. (Example of degeneracy traced in detail for Vietnam.) This is (along with the large number of places whose latitudes fall conspicuously upon exact klimata) one of the strongest arguments for the DR theory that rounded longest-day data (GD 8's or its type) were the basis for the data of GD 2-7.
    Note the historically important (if paradigmist-verboten DIO 2.1 [1992] ‡3 §C10 [p.31]) lesson imparted: competent ancient geography's heritage to us was severely corrupted — crippled is a more accurate indictment — by the societal ubiquity of a pseudo-science, astrology. But keep in mind (DIO 4.3 ‡15 §C3 [p.124]) that Ptolemy worked for the newly-cosmopolitan, astrology-saturated Serapic religion, and doing horoscopes internationally requires (then & now) 3 manuals: astronomical tables, geographical tables, & interpretational handbook. Ptolemy's prime works were: Almajest, GD, & Tetrabiblos.

  • Suggested Solution to Two Mysteries:
    As shown in the tables of Rawlins Vistas 1985 p.262, GD latitude-errors for major cities are often sph-trigonometrically consistent with the DR theory. See op cit p.261, for the relevant math. See also discussion (ibid p.259) of a further revealing point: without the DR theory presented there & here, how could one reasonably explain two shocking oddities (which had evidently escaped the notice of previous commentators):
    [1] GD latitudes (as already noted) are two ordmags cruder than ancient astronomers' latitude-accuracy. (Roughly: a degree vs an arcmin.)
    [2] The GD latitude errors' large size (again: ordmag a degree) is comparable to that of its longitude errors — this, though:
    [a] The former should be 30 times smaller than the latter. (Or 41 times smaller, if eclipse-observations aren't taken as raw-data pairs.)
    [b] Real astronomers knew their latitude to ordmag an arcmin. (D.Rawlins Isis 73:259-265 [1982b] p.263 n.17.)
    [Note that GD 1.2 shows awareness that astronomical observation is the most reliable basis of latitude-measure. This returns us to the question: if sophisticated cities knew their latitude, how did most of these data get corrupted by astrologers? Was there a long astrological tradition of geographical tables, which Marinos (note GD 1.17.2's semi-connexion of astrologers' klimata to Marinos) and-or Ptolemy felt forced to assent to the flawed important-cities latitudes of? (Just as usually-equant-preferring Ptolemy may've felt forced to go along [in the Almajest] with Hipparchos' flawed but long-pagan-sacred eccentric-model solar tables.)]

    GD 8's Disconnect:
    The GD a Hybrid

  • The order of data-listing for GD 2-7 and GD 8 are similar. (And the former's 26 local maps correlate in designation and sequence with the latter's.) This suggests some sort of inter-causation or co-causation. (GD 8.2.1's statement that GD 8's data are from degree-lists does not say that they were those of GD 2-7, though that may be the implication and-or the truth.)

  • However, throughout the GD, we find repeated instances of differences in order-of-listing. [E.g., Nîmes (D29) & Vienne (D28): GD 2.10.10-11 & 8.5.7. (Berggren & Jones 2000 p.106 vs p.122.) Kasandreia (D101) & Thessalonike (D95): GD 3.13-14 & G8.12.10&4. Pergamon (D178) & neighborhood: GD 5.2.14 & 8.17.10. Hierapolis (D237) & Antioch (D236): GD 5.20.13&16 & 8.20.8&7. Teredon (D259) & Babylon (D256): GD 5.20.5&6 & GD 8.20.30&27. Kattigara (D356) & Thinai (D355): GD 7.3.3&6 & GD 8.27.14&12.]
    Which argues against GD 8 being computed directly from GD 2-7 or vice-versa.
    [For the consistent sites, either there were calculations of one section's data from the other (in one or both directions) or scrupulous attention was paid to math-consistency between the two sections (whether at the outset or during later editors' touchings-up) — though there are occasional inconsistencies, e.g., the longitude of Rome (D49): GD 3.1.61 puts Rome 36°2/3 west of the Fortunate Isles, while GD 8.8.3 puts Rome 1h5/8 east of Alexandria. (Itself 60°1/2 west of Blest Isles by GD 4.5.9, or 4h [60°] by GD 8.15.10. See DR 1985 n.25.) But (60°1/2 − 36°2/3)/(15°/hour) ≈ 1h7/12 < 1h5/8. Similar incompatibility: Salinae (GD 3.8.7, 8.11.4, D79).]

  • Decades ago, Aubrey Diller pointed out to DR that the GD never mentions Book 8 — until the reader arrives there.

  • DR has noted something similar: throughout GD 1, there is no mention of Alexandria, Ptolemy's claimed home and his Almajest's prime meridian.
    [Nobbe 1:46 inserts Alexandria at the 14h klima (GD 1.23.9), but it is clear from C.Müller (1883) 1:57, Berggren & Jones 2000 pp.85&111, and Stückelberger-Graßhoff-Mittenhuber 1:116 n.4 that this was not in the original, which (in GD 1.23) named only four klimata north of the Equator: Meroë [D165] (13h), Syene [D154] (13h1/2), Rhodos [D189] (14h1/2), Thule [D1] (20h). Selection repeated GD 7.5: Berggren & Jones 2000 p.111. Note that Alexandria [D149] is mentioned at GD 7.5.13-14.]
    By contrast, GD 1 mentions such sites as: Thule (D1), Ravenna (D56), Lilybaeum (D67), Carthage (D131), Rhodos (D189), Canopus (Ptolemy's true home), Syene (D154), Meroë (D165), Arbela (D261), Okelis (D281), Kattigara (D356), among many others. Since Ptolemy is a multiply-convicted plagiarist (DIO 12 [2002]), one may ask: is it credible that allegedly (Almajest 3.1) Alexandrian Ptolemy would write a preface to his Geography which never mentions his own city, when it is the prime meridian for his astronomical works, for his earlier-announced (Almajest 2.13) forthcoming geography, and for GD 8?

    Blest Isles Ignored & Identified: the Cape Verde Islands

  • Conversely, the Blest Isles, the GD ekumene's west bound (and GD 2-7's implicit prime meridian), have no entry in GD 8. In GD 8, this linch-pin site is only mentioned at two places, rather in-passing: at GD 8's prime meridian Alexandria (GD 8.15.10) and at the GD ekumene's east bound, Thinai (GD 8.27.13), where it is noted that Thinai is 8h east of Alexandria and thus 12h east of the Blest Isles.

  • Yet another oddity: the GD repeatedly states that the Blest Isles are the west bound of the ekumene. (Though, curiously, not at GD 7.5.2, even while soon after saying so at GD 7.5.14.) Yet the writer of GD 1 does not explicitly state that all the longitudes of GD 2-7 will be measured from the Blest Isles; and the Blest Isles' has no entry in GD 8. Its position appears under Africa at GD 4.6.34.
    [Thanks to Alex Jones for pointing this out.]

  • Additionally, one notes that there is not a single absolute longitude in GD 1 — every longitudinal value is given in strictly differential terms. Now, if one is writing a preface to a compendium that provides the longitude-east-of-Blest-Isles of 8000 sites, one would think that the east-of-BI part just might get mentioned somewhere. Instead, GD 1 is completely non-committal regarding what will be the prime meridian of the work. And GD 2.1 (the preface that launches the reader into the 8000 sites) is likewise.
    (If one were just grabbing — virtually unedited — a preface to another work, something like this could easily happen.)

  • In the GD, there are a few islands near Mauretania at about the latitude of the Canaries, which are the hitherto-standard identification of Ptolemy's Blest Isles. (E.g., Stückelberger & Graßhoff 2006 1:455 n.200, which scrupulously notes that the identification of the Blest Isles with the Canaries is uncertain.) But these islands are not GD-listed at longitude zero; nor is the center of the real Canaries longitudinally beyond the real western hump of Africa, which is how the western-most anciently-known land obviously ought to lie.

  • GD 4.1.16 lists a few non-zero-longitude islands (Paina & Erytheia) at latitude c.30°N and c.3° west of Mauritania, which would be the Canaries. But these are roughly a thousand miles north of Ptolemy's six “Blest Isles” which are listed by him at GD 4.6.34, at longitudes 0° (four) or 1° (two), at north latitudes ranging from 10°1/2 to 16° — which is about right for the Cape Verde Islands. (Actual CVI latitudes: c.75 nmi N&S of 16°N.) The same isles are also visible on GD maps (see, e.g., Berggren & Jones 2000 plate 6 [from c.1300 AD]; same location in plate 1; Stückelberger & Graßhoff 2006 2:838 and volumes' inside-covers) marked as the Blest Isles (“Fortuna insula”), strung along a longitude of about 0° — 300 nmi west of the western-most point (hump) of Africa (Dakar, on Cap Vert) at a latitude that is again a convincing match for the Cape Verde Islands, which are therefore firmly identified as the Blest Isles. (Online, the same six “Fortunate” islands can be seen at the west end of Ptolemy's world map, again at a position pretty consistent with that of the Cape Verde Islands.) The persistent previous confusion presumably occurred because the 5th of the 6 islands listed at GD 4.6.34 is called “Kanaria Nesos”.

  • The GD's knowledge of the Cape Verde Islands stands as a testament to ancient explorers' courage: they are c.300 nmi from the mainland. (By contrast, the Canaries are barely off the NW-Africa shore.) So the islands' discoverer was himself the nearest thing to an ancient Eriksen or Columbus. More than a millennium before sailors discovered tacking, trips there were presumably extremely rare and hazardous. Possibly galley-slave rowing-power was the key to the ancients' knowledge of the Cape Verde Islands. And perhaps they were regarded as Blest because European civilization had not yet significantly uplifted the inhabitants by the introduction of its ever-brewing wars & their ever-resultant slavery.

    Hours as the Route of All Evil in Ptolemy's GD

  • Looking at GD 1-7 and GD 8 as separate sections of the GD, one must notice that each of the two sections' cross-citations of the other's prime meridian is paltry at best (and could well have been from later interpolation) — so let's keep our eye on the main point: there is no mention of the Blest Isles in the preface to GD 8, any more than there is any mention of Alexandria in the preface (GD 1) of GD 1-7. It would be hard to ask for better evidence that neither section was the immediate direct source of the other's totality.

  • But let us return to the essence of the DR theory that the major-site data of GD 2-7 were based upon data of the type found in GD 8, and fix upon the main points regarding the source of GD 2-7's data:
    [a] Whereas all latitudes were originally measured angles (method: Almajest 1.12), the inaccuracy of the latitudes in GD 2-7 show that these data had been corrupted by subjection to crude rounding for astrologers' longest-day tables in hours, before being computationally converted into the latitude-degree data that ended up in GD 2-7.
    [b] All astronomically-based longitudes in GD 2-7 were originally in hours, as noted in GD 1.4.
    [Wrongly, Ptolemy believed (GD 1.4&12-13) that eclipse-based longitudes were rare.
    (The method of finding longitude-differences between sites by comparing local times of simultaneously-observed lunar eclipses, was obviously well known. See, e.g., Strabo 1.1.12 or GD 1.4.2. Least-squares tests on ancient longitudes show that the eclipse method had been extensively used by genuine ancient scientists: Rawlins 1985 §§5&9 [pp.258-259 & 264-265].)
    And so he assumed that generally-accepted longitudes were primarily based upon travellers' stade-measured distances (terrestrial) instead of eclipse-comparisons (celestial) — a crucial, disastrous error, which undid generations of competent scientists' eclipse-based accurate longitudes-in-hours and thereby wrecked the GD's longitude macro-accuracy in angle. (Though not in distance.) Note: said mis-step must have occurred before the hypothetical dovetailing of GD 2-7 and GD 8, perhaps in the 1st century BC.]

    This, because based upon comparisons of lunar-eclipse local-times.
    [A number of network-cities' GD 2-7 longitudes could have been calculated directly from GD 8 or its source, using Alexandria (D149) longitude (east-of-Blest-Isles) 60°1/2 (GD 4.5.9) or 60° (GD 8.15.10). Some examples:
    London (GD 2.3.27, 8.3.6, D4), Bordeaux (2.7.8, 8.5.4, D21), Marseilles (2.10.8, 8.5.7, D26), Tarentum [Diller (DIO 5 [2006]) Codices XZ Europe-Map 6 site #5] (3.1.12, 8.8.4, D53), Brindisi (3.1.13, 8.8.4, D54), Lilybaeum (3.4.5, 8.9.4, D67), Syracuse (3.4.9, 8.9.4, D68), Kyrene (4.4.11, 8.15.7, D146), Meroë (4.8.21, 8.16.9, D165), Kyzikos (5.2.2, 8.17.8, D176), Miletos (5.2.9, 8.17.13, D181), Knidos (5.2.10, 8.17.14, D182), Rhodos (5.2.34, 8.17.21, D189 — allowing for common [DIO 4.1 [1994] ‡3 §F3 [p.43] ] ancient rounding of 1h/8 to 8m), Jerusalem (5.16.8, 8.20.18, D247), Persepolis (6.4.4, 8.21.13, D271).
    However, these could as easily have been computed in the other direction. The majority of less grid-critical sites' degree-coordinates couldn't have been computed directly from those of GD 8 (at least in its present state), but could've gone the other way; e.g., Smyrna (5.2,7, 8.17.11, D179) & Pergamon (5.2.14, 8.17.10, D178).
    Given the GD as it stands, if GD 8 is contended to be the direct ancestor of GD 2-7's longitudes, one would have to argue that the underlying network-basis was far less in number than GD 8's 360 sites — which, if we are speaking of sites whose longitudes (vs Alexandria) had been astronomically determined, would not (in itself) be an unreasonable contention.]

    [c] Thus we have arrived at a hitherto-unappreciated realization: ironically, every jot of the astronomically-determined data of the basic network of cities underlying GD 2-7's thousands of degree-expressed positions, was at some point (during its mathematical descent from its empirical base) rendered in time-units: hours. As proposed in DR 1985.

  • And, as a result of rounded longest-days and Earth-scale shifting, these hour-data became the semi-competent-occultist conduit for data-corruption which tragically destroyed a sophisticated civilization's laboriously accumulated high-quality astronomically-based ancient geographical data.

    Precession and Aristarchos

  • Precession is the difference in the length of the tropical and sidereal year, caused by a gradual shift of the Earth's axis — an ancient discovery which we can easily trace back to Aristarchos (not-so-coincidentally also the 1st astronomer to publicly announce that the Earth moved), since he is the earliest ancient cited to two different year-lengths.
    [Note: not a single historian has yet indicated publicly that he understands this rather self-evident point. (Though some have privately.) Which gives us hope that sociology can yet attain to the predictivity of astronomy.]
    Aristarchos flourished c.280 BC: 1 1/2 centuries before Hipparchos, hitherto generally regarded as precession's discoverer. Both of Aristarchos' yearlengths are provided at DIO 9.1 [1999] ‡3 §B7 [p.33]; see also DIO 11.1 [2002] ‡1 nn.14&16 [p.8].

  • Precession was known to the author of GD 8.2.3. [GD 8.2.2 by the arrangement of L.Berggren & A.Jones (Princeton 2000); or 8.B.2 in Aubrey Diller's 1984 translation at DIO 5 [2006].] Thus, the GD 1.7.4 discussion seems awfully strange (though some experts disagree: Berggren & Jones 2000 p.65 n.23 & p.120 n.3), since it here quotes the statement of Marinos of Tyre (c.140 AD) that all the constellations rise&set in the tropical geographical regions — with the sole exception of UMi, which becomes ever-visible after a northward traveler passes north latitude +12°2/5 , Hipparchos' long-precessionally-obsolete NPD (North Polar Distance = compliment of decl) for αUMi. (I.e., modern “Polaris”: the brightest star in UMi, and the most northern easily-visible UMi star for us; the most southern for Hipparchos.) And αUMi's NPD actually was 12°27' (Decl = 77°33') at Hipparchos' chosen epoch, −126.278 (128 BC Sept.24 Rhodos Apparent Noon: DIO 1.1 [1991] ‡6 eq.28 [p.58]). Marinos further states that this parallel is 1° north of Okelis, which he mis-places at 11°2/5 N latitude. (A poor estimate, since Okelis (D281) [modern Turbah, Yemen] is actually at 12°41'N, 43°32'E.)
    [Is this a revised&multibungled re-hash of an original Hipparchos estimate that Okelis was on the arctic (ever-visible) circle of αUMi? — which would have been correct in 170 BC and OK to ordmag 0°.1 during his career.]
    Yet, by Marinos' time, αUMi's NPD had precessed down to about 11°: in 140 AD, it was 10°59'. So, his statements prove he didn't account for precession.
    But the most peculiar aspect of this matter is that GD 1.7.4 makes no comment at all on Marinos' flagrant omission of precession — and this though Ptolemy is (as usual) in full critical mode (alertly questioning [GD 1.7.5] whether any of Marinos' discussion is based upon the slightest empirical research), and though the writer of the Almajest certainly knew (Alm  7.2-3) the math of precession. Comments:

  • There can be little doubt that the authors of GD 1.7.4 and GD 8.2.3 were not the same person.

  • If Okelis were where Marinos placed it, αUMi's ever-visible circle would have been south, not north of Okelis.

  • Has it been noted that, by the time of Marinos & Ptolemy, αUMi was (thanks to precession) no longer the most southern of UMi's seven traditional stars?! — ηUMi and especially 3rd magnitude γUMi were much more so.
    Indeed, for the time of the GD, γUMi was over a degree (1°04' at 160 AD) more southern than αUMi. (Shouldn't the “Greatest Astronomer of Antiquity” [Science 1976/8/6] have known this? — especially since he pretended he'd cataloged the whole sky's stars: Almajest 7.4. I.e., the GD 1.7.4 statement on αUMi disagrees not only with the sky but with Ptolemy's own tables.) Thus, γUMi had long since assumed the distinction (one interjected by Marinos, ironically) of being the outrider-star whose NPD determined whether a geographical region was far enough north to attain UMi-ever-visibility. (Note that GD 6.7.7 puts Okelis at latitude 12°N [and false-Okelis at 12°1/2]; so, creditably, the GD's Okelis latitude was closer to reality than to Marinos. Note also that 12° is almost exactly the theoretical ever-visible latitude for all of UMi at the GD's epoch, since γUMi's NPD was 11°56'+ in 160 AD.)
    [Likewise, 1000 nmi to the southwest of Okelis: regarding the location of the two lakes feeding the Nile, the GD astutely makes a major correction to Marinos in placing both lakes much nearer the Equator than Marinos had them. (In reality: the Equator runs through the eastern source, Lake Victoria. And the western source, lake-pair Edward & Albert, straddles the Equator.) Remarkably, the GD's maps of Africa were still consulted by geographers in the mid-19th century, when these lakes were finally 1st reached by Englishmen. (See J Roy Geogr Soc 29:283, 35:1, 7, 12-14; Proc RGS 10:258.)]
    As noted, the foregoing strongly suggests that the same person did not write GD 1.7.4 and GD 8.2.3. And several other features suggest independently that the GD is a patch-work opus: see D.Rawlins Vistas in Astronomy 28 [1985] p.260 [On vs Heliopolis]; and p.266 & n.6.
    [We find similar hints of patch-workery throughout the GD, e.g., at GD 1.24.11-vs-17, as the lettering for two consecutive projection-diagrams are needlessly shuffled. (See Berggren & Jones 2000 p.91 n.80.) See also another Ptolemy-compiled work, the Almajest, where, e.g., the mean motion tables' Saturn→Mercury order of the planets (Alm 9.3-4) is the reverse of the Mercury→Saturn order followed in their fraudulently (Rawlins Amer J. Phys. 55 pp.236-237 item 5; DIO 11.2 [2003] §K [pp.48-49]) alleged derivation at Alm 9.6-11.8. For more such patch-work indications, see frequently here, and at DIO 8 [1998] ‡1 end-note 17 [p.17] & DIO 11.3 [2002] ‡6 §C [p.76].]
    Thus, the above analysis of GD 1.7.4 provides another powerful augmentation of that evidential collection.
    [Indicia of such patch-workery in the GD are frequently noted here, due to the inexplicably-repeated modern claim of coherent unity for each of Ptolemy's works.]

    Marinos Mis-Dated?

  • Nowadays, it seems to be almost universally assumed (e.g., Neugebauer 1975 pp.879&939) that Marinos flourished very early in the 2nd century AD, sometime during Trajan's reign, around 110 AD.
    [Quite aside from the present discussion: for compelling evidence against this date, see H.Müller's clever discovery.]
    Which is curious, since in c.160 AD (or perhaps even later: DIO 4.1 [1994] ‡3 n.45 [p.45]) Ptolemy refers to Marinos as (GD 1.6.1 emph added): “the most recent [of those] of our time” who have attempted a large geography. Now, if you were currently writing of a geographer of the mid-1950s, would you speak of him so? (GD 1.17.1 has been taken to indicate that Marinos was retired or dead by Ptolemy's day, but the passage is hardly unambiguous on that point — and would make more sense if Marinos' latest publication was merely 5 or 10 years past.)

  • Moreover, Alex Jones points out (2007/5/23 conversation) that the forward dating of Marinos would help solve a problem first emphasized by Paul Schnabel (Sitzungsberichte der preussischen Akademie der Wissenschaften 1930 phil-hist Klasse XIV:214-250 p.216): when did Ptolemy become aware that people lived south of the Equator? Almajest 2.6 says the S.Hemisphere is unexplored, though Marinos says otherwise and the GD agrees.
    This implies, since the Alma  might've been compiled during Marcus Aurelius' reign, that Marinos' date could be as late as c.160AD.

  • The argument adduced to date Marinos to much earlier (than Ptolemy) is that Marinos' work took into account names of sites reflecting the changing Empire, e.g., Trajan in Dacia (GD 3.8, 8.11.4 [roughly modern Romania]) up to c.110 — but not later in Parthia (GD 6.5, 8.21.16-18 [roughly modern Iran]) and north Africa. But how sure is such tenuous reasoning? How strongly should it rank? — in the face of:
    [a] GD 1.6.1's plain statement of Marinos' contemporariness, and
    [b] the incredibility of the long-orthodox implicit assumption that, in a busy mercantile empire, a succession of macro-geographers (GD 1.6.1 implies plurality) suddenly ceased for 1/2 a century!

  • Moreover, why assume that Marinos adopted all the latest name-changes? Ptolemy didn't: his preface's criticisms complain (GD 1.17.4) that Marinos misplaced the Indian trading town Simylla (D330) and didn't realize that the natives call it Timoula. Yet the GD's data-listings (GD 7.1.6 & 8.26.3) both retain Marinos' name: Simylla, not Timoula. Berggren & Jones 2000 n.53 (p.76) note an even more revealing careless retention: Marinos' Aromata latitude.
    [These situations remind one of the common modern mis-interpretation (DIO 11.1 [2002] ‡2 n.7 [p.12]) of Almajest 3.7 to mean that no Babylonian astronomical records came through to Ptolemy prior to 747 BC, though the actual statement is rather that continuous records went back that far.]
    So, what should be tested isn't whether all but whether any post-Trajan geography appears in the GD.

  • Especially since it doesn't seem that there'd likely be many changes. After all, it's well-known that Dacia was the last solid addition to the Roman Empire. (It may not be coincidental that around this time the Roman army was becoming predominantly alien-mercenary.) Trajan's army was of course stronger than Dacia's. (So, we know who ended up with Dacia's gold.)
    [Trajan aureus (possibly containing said gold): Kunsthistorisches Museum, Vienna.]
    But it wasn't stronger than that of the Parthian Empire; thus, the attempted-rape victim got in all the Part'n shots, and the puppet ruler whom Trajan had placed into power at the then-capital (Ctesiphon [D262], near Babylon [D256]) passed on soon after, as did Trajan (117 AD).
    [Over 4 centuries of botheration, Parthia repelled three Roman invasions: swallowing the army of Crassus (suppressor & crucifier of Spartacus, and member of the 1st triumvirate), exhausting emperor Trajan and (after a temporary setback at Marcus Aurelius' hands) slaying last pagan emperor Julian the Apostate. And, yes, “parting shot” is thought to come from Parthian archers' tactic of shooting arrows even when retreating or pseudo-retreating.]
    Trajan's adventure in Parthia having been an expensive failure, his two successors chose not to try expanding the empire. Hadrian (117-138) did not share certain current warlords' fiscal profligacy. Similarly for Antoninus Pius (138-161: which takes us up to the time of Ptolemy's geographical work).
    [Antoninus Pius aureus: DIO Collection.]
    These points urge some caution before we draw conclusions on Marinos' date from lack of the-very-latest Parthian information.

  • Next, we note that the most notorious exception to the non-expansion policy of Hadrian occurred in Palestine. In 130 AD, he visited Jerusalem and ordered its re-building. Since Hadrian's family name was Aelius, he re-named Jerusalem: “Aelia Capitolina”. (His supervision evidently triggered a local revolt — put down in 132-134, with Hadrian sometimes on the scene.) So, does the GD reflect the change? Yes: GD 5.16.8 lists “Ierosoluma [Jerusalem], which is called Ailia Kapitolias”. And GD 8.20.18 lists “Ailia Kapitolias Ierosoluma” without further comment but obviously reflecting the same up-to-date information.
    [Such an explicit update is rare in the GD's data-body. Another such passage, even more unusually discursive, is found at GD 7.4.1, where it is stated that Taprobane (modern Sri Lanka [though known as Ceylon in Diller's & DR's youth]) was formerly called Simoundou but is now called Salike by the natives. Comments are even (very atypically) added, describing Salike's women and local products ranging from meal & gold to elephants & tigers. It seems likely that the mention of both Ailia Kapitolias and Salike were late additions to the GD, a point we will later make use of.] (Note: Taprobane [GD 8.28] is the last map in the GD, though [given its location] it should obviously have been covered before the GD listings get to China. I.e., we have here yet another symptom of a late add-on.)
    Therefore, we have indication that both the GD's data-sections (GD 2-7 and GD 8), previously adduced to date Marinos to c.110, actually contain material from the 130s or later.
    [Following the revolt's suppression, Judaea was re-named “Syria Palestine” and Rome henceforth (c.135) eliminated the term “Judaea”. The fact that it is retained in both the body (G2-G7) and Book 8 of the GD, taken together with the re-naming of Jerusalem leaves us with a bracket-argument in favor of dating Marinos to about 135, which is indeed of Ptolemy's time — as he said.]

  • An example of the fruitfulness of the foregoing:
    Almost 2 centuries ago, H.Müller made the brilliant observation that a GD-listed N.German town “Siatoutanda”, was probably non-existent — just (another) Ptolemy-compilation mis-read of a foreign language: Tacitus' Latin description (Annals 4.73) of a N.German battle-retreat (“ad sua tutanda”).
    [This does not stop our ancient geographer from providing highly specific coordinates: longitude 29°1/3, latitude 54°1/3 (GD 2.11.27).
    Reminds one of St.Philomena, of whose “life” whole books used to be written (DR possesses a copy of one), though she never existed: “Philomena” turned out to be just an over-imaginative later mis-read of a fragmentary ancient stone inscription (found in the catacombs of Rome on 1802/5/25): “LUMEN PAX TECUM FI”, which was “restored” as a reference to FILUMEN or Philomena. This was enough to launch (starting c.1805 in the super-religious Kingdom of Naples) a cult, special novenas, the usual “miracles”, and (from devotees' revelations) a detailed biography of her life & martyrdom. The Roman church creditably removed her from the list of saints about a 1/2 century ago.]

    As is all-too usual in the ancient-science community, Müller's novel and obviously valid discovery has been doubted on grounds so tenuous (in comparison to the compelling evidence in its favor) as to make one wonder whether anything ever gets resolved in this field, no matter the power of relative evidence.
    [What does that say about the field? See DIO 7.1 [1997] ‡5 n.40 [p.33]. Note the Velikovskian context.]
    Against Müller, it has been argued (see sources cited at Berggren & Jones 2000 p.28 n.34) that the Tacitus work was published in 116 AD, which is after the (inexplicably-widely-believed) upper-limit date (110 AD) for Marinos. But the 110 date is so far from firmly established that one should reverse the situation: instead of using the date to exclude H.Müller's finding, use the HM finding to help establish a lower limit for Marinos' date.
    [Similarly, when (1999/10/1) dim atmosphere proponent B.Schaefer imparted to DR his intention of testing the Ancient Star Catalog's authorship by assuming 0.23 mags/atm opacity, DR immediately suggested that it would be far more fruitful to use Hipparchos' authorship (which had by then been obvious to serious astronomers for centuries) to test for ancient atmospheric opacity. BS didn't listen, so this important and revealing project — proving beyond any question that man (not nature) is the prime cause of present atmospheric opacities ominously higher than ancient skies' — was instead masterfully and independently established by Keith Pickering in DIO 12 [2002] ‡1 §§D2-D5 [pp.11-12].]
    So we recognize that H.Müller's discovery contributes importantly to the evidence suggesting that conventional wisdom on Marinos' date is suspect, and thus that there is little trustworthy evidence against our proposal that Marinos was much nearer Ptolemy's contemporary than is now generally understood.

    Tyre: Missing Home-City of Book 8's Once-Supposed Source

  • The most peculiar coincidence in the history of ancient geography will turn out to be a lucky break for scholars of the GD: incredibly, Marinos' native Tyre is absent from GD 8.
    Tyre's absence from GD 8 has several non-neatnesses. While Tyre is also missing from the Important Cities lists in late copies of Ptolemy's Handy Tables (Halma ed.), Tyre does reside in two 9th century copies (published by Honigmann), which are far older than our earliest mss of the GD, and each contains c.100 more sites (than GD 8): Tyre is city #307 in Vat 1291, #160a in Leid.LXXVIII. In the latter ms, Tyre is counted secondarily; which suggests that, if paring occurred, Tyre was expendable. The superficially attractive interpretation is to wonder if GD 8 is a Byzantine-era add-on, which reflected a shrinking of the number of sites from nearly 500 to just 360.
    The problem with that theory is format: GD 8 differs generically (from all other surviving Important Cities lists, which uniformly are in longitude degrees east of the Blest Isles and latitude degrees north of the Equator) by: [1] using Alexandria as prime meridian (astrologer Ptolemy's preference); and [2] providing data entirely in hours, just as astrologers prefer. This argues strongly that GD 8 goes back in time at least as far as Ptolemy.]
    Curiously, this telling point has been overlooked in the literature. And, in a context of questionable authorship, we must notice that, while Marinos' home-city (Tyre) is missing from GD 8, Ptolemy's alleged home-city (Alexandria) is likewise missing from GD 8.

  • Marinos is clearly identified as of-Tyre (GD 1.6.1). Indeed, Tyre (Phoenicia) is cited doubly and with accurate latitude — highly exceptional on each count — at GD 5.15.5&27:  67° E of Blest Isles, 33°1/3 N of Equator. (The latitude is correct if we account for refraction of pole-star light and 5' rounding.)

  • Thus, we conclude that GD 8 (in the form we have it) was not compiled by Marinos.

    Landlubber Ho!
    Ancient Saigon & Hanoi
    Wrapped China Negates the Pacific

  • It is well-known that the farthest-east region of the GD, China, portrays a non-existent continuous roughly-north-to-south coast (blocking any route to the Pacific) beyond the South China Sea, near longitude 180° (12 hours) east of the Blest Isles or 120° (8 hours) east of Alexandria, stretching from near the Tropic of Cancer, all the way south to Kattigara at 8°1/2 S. latitude — effectively wrapping China around the Indian Ocean's eastern outlet. The latitude-longitude coordinates for eighteen China sites are found in GD 7.3 (Renou op cit p.62-65).

  • But, according to the previously-broached DR theory, all of this geography hinges upon the underlying grid-network: GD 8 and-or its kin. If we look at the GD 8.27.11-14 China data, we find that the situation of all China hinges upon just three cities' hour-data (longest day and longitude east of Alexandria, according to Diller's XZ-tradition mss): Aspithra [D354] (13h1/8, 7h2/3), Thinai [D355] (12h5/8, 8h), Kattigara [D356] (12h1/2, 7h3/4). Anything wrong with GD's China is wrong in this trio. And trouble won't be hard to find.

  • For Thinai (D355), GD 7.3.6's latitude (3°S) jars with GD 8.27.12's longest-day 12h3/4 north, which would be correct for about latitude 12°1/2 N.

  • Fortunately, Vat 1291's Important Cities compendium lists the same three cities (only) for China. (Honigmann op cit p.206: cities #443-#445; no China listings in Leid.LXXVIII.) And on Thinai, it provides confirmation of GD 8 (not GD 7), listing Thinai at 13°N. Which suggests that the 3°S of GD 7 is either a scribal error (missing the iota for ten) or perhaps is differential: 3° south of Aspithra (16°1/4N). Either way, it seems that 13°N is correct, as listed by Vat 1291. (Stückelberger-Graßhoff-Mittenhuber 2:734 for Thinai has GD 7.3.6's 13°latitude.)
    [The same Vat 1291 list gives 18°1/4 N latitude for Aspithra (not the 16°1/4 N latitude of GD 7.3.2, corresponding to longest-day 13h1/8, the very Aspithra longest-day value listed in Diller's XZ-tradition mss. (One is tempted to ask if 18°1/4 latitude was the true original latitude — or was later forced to agree with M = 13h1/8? But it could have just come from a scribal error.) In Nobbe, GD 8 lists Aspithra at longest-day “about” 13h, which corresponds to latitude 16°+ — agreeing with the GD 7.3.5 Aspithra latitude in Nobbe and Renou: 16° and 16°1/4 N, respectively.]

  • Finally, we observe that Kattigara's latitude in degrees is the same in both Vat 1291 and GD 7.3.3 — but in the former it is north latitude (which makes way more sense for a Chinese city), correctly contradicting the impossible southern latitude of both GD 7.3.3 & GD 8.27.14. The matter gets even more interesting when we check our corrected position for Kattigara: 177°W (of the Blest Isles) & 8°1/2 N — that is precisely the GD 7.3.2 position of Rhabana. Therefore (not for the 1st time: see Rawlins 1985), the GD may have used two (or more) names for the same place.

  • Thus, when we examine the underlying-grid trio for China, the two negative (southern) latitudes both appear so shaky that we can dispense with all negative signs for China — which eliminates the above-cited fantastic N-S coastal-bar to the Pacific.

  • There is a disturbing pattern to the GD 7 latitudes of the only four S.China Sea cities on the Vietnam coast which are listed in GD 8 (in order N-to-S): Aspithra, Thinai, Kattigara, Zabai. These cities' GD 7.2-3 latitudes are, resp, about equal to: 16°1/4, 13°, 8°1/2, 4°3/4 — which are suspiciously close to what one would compute indoors via sph trig from a quarter-hour-interval klimata table: Aspithra (D354) 13h, Thinai (D355) 12h3/4, Kattigara (D356) 12h1/2, Zabai (D348) 12h1/4. (And, indeed, these are the values Diller found in GD 8's UNK mss-tradition.) This looks even fishier when one recalls (above) that these are the only 4 cities on the Vietnam coast which are listed in GD 8, where only longest-days (the stuff of klimata-tables) are provided for N-S position. (Even the precise 13h1/8 variant discussed above for Aspithra, perfectly matched what may have been merely a scribal error: 18°1/4.) Obviously assuming exactly-correct latitudes here is risky when dealing with such rounded data. Conclusion: we must also use verbal descriptions, if we wish to have any chance of solving this section of the GD.

  • So, where and what are these cities? GD 7.3.3 refers to Kattigara (which has a 1st syllable like Cathay's) as a Chinese harbor, near walled cities and mountains. So there's little doubt that we are on the Asian mainland, in the area of Vietnam-China.
    [Note: The rest of this explicitly speculative reconstruction was nontrivially re-analysed & revised in 2009. See DIO 5 [2009] n.68 [pp.15-16] for numerous SE Asia site-identifications.]
    Our interpretation of GD 1.13.9 (B&J p.75): Marinos is saying that an ancient sea voyage from Malay's Sabara-Tamala region (Phuket, Malay) to the Golden Peninsula (Sumatra's NW tip) is roughly 200 mi, which is about right. (Marinos' sailing direction [c.SE] is ignored here, since based on his distorted map.) GD 1.14 says the rest of the trip to Zabai (Singapore) takes 20 days. Going around Sumatra (instead of sailing between Malay&Sumatra) would require c.20 days. (Speed c.100mi/day: already established at B&J p.76 via GD 1.14.4: Aromata to Prason. Made more exact by checking Phuket-to-Singapore.) The original report is due to “Alexandros” (geographer? explorer?) who says the trip from Zabai across to Kattigara (Saigon) takes merely “some days” (GD 1.14.1-3), roughly consistent with the c.6 days it would've taken at the previous speed.

  • The GD's supposed direction to Kattigara (left [east] of south) is obviously confused. I suspect that the ancient cause was a common land-lubber misinterpretation: “south wind” (which means wind from the south) was taken as towards the south — thus, the report of going somewhat east of a “south wind” (GD 1.14.1; B&J p.75) was mis-taken (at GD 1.14.6; compare to B&J p.76) to mean sailing with a wind blowing southward.
    [Would linguistic problems (in the babel of antiquity) have contributed to these errors? (Marinos likely wrote in Greek; otherwise, Ptolemy could not have used him for a whole book.) Again, for Ptolemy, it probably wouldn't have been the 1st time. He appears to have sloppily misordered (GD1.4.2) simple, well-known data regarding the famous lunar eclipse that occurred shortly before the Battle of Arbela (D261 [modern Irbil, lately a north Iraq hot-spot]) also seen at Carthage (D131), by screwing-up the Latin text of (or like) Pliny's accurate description of that −330/9/20 event, thereby attaching Arbela's eclipse-time to Carthage! Despite lunar eclipse after lunar eclipse occurring in Ptolemy's lifetime (three recorded at Alexandria in under 3 years at Almajest 4.6: 133-136 AD), this antique record was his sole example (!) of how to determine longitude astronomically. Further suggestion of patch-workery: the Ptolemy account of these eclipses is in gross disagreement with not just the real sky but just as grossly with his own luni-solar tables. See similar situations for Polaris at DIO 14 [2008] ‡3 n.31 [p.43] and for Venus at DIO 11.3 [2002] ‡6 §B3 [p.74]. And his solar fakes also show the same propensity to swift-simple, not-even-tabular fraud and plagiarism. (Anyone researching Ptolemy should keep ever in mind that he was shamelessly capable of every brand of deceit. See DIO 8 [1998] ‡1 end-note 2 [p.14].) This eclipse was so famous that one would suppose it was widely-written-of. Thus, it is doubly weird that Ptolemy could make such an error. The suggestion here is that, as an astrologer for a Serapic temple, he was isolated from real scientists. (As perhaps Hipparchos had also been.)]

  • Kattigara (D356) was probably about where resides the harbor long called Saigon. (Re-named Ho Chi Minh City. For now.) The real Saigon's latitude is just north of 10°N, so the GD is off by c.2°, which is about as big an error as one will find caused in this region by computing latitudes from 1h/4-interval klimata. Whoever originally cubby-holed Saigon so found that its L didn't fall exactly on a klima: the nearest such klima for rounded L = 10° would in a region rounding to 1°/4 put L at 8°1/2. This, in microcosm, is the secret of why the GD's mean latitude error is so poor: ordmag 1°, despite contemporary astronomers' achievement of knowing their latitudes ordmag 100 times more accurately. (See citations: Rawlins 1982a, 1982b, & 1985).

  • For the four above-cited SE Asia cities with klima-afflicted latitudes, our tentative identifications follow.
    Inland Aspithra (D354, L: 16°1/4) = Thailand Gulf's Chanthaburi (real L: 12°.7).
    Deeply inland Thinai (D355, L: 13°) = Cambodia's Phnom Penh (real L: 11°.6).
    Kattigara (D356, L: 8°1/2) = Saigon (real L: 10°.8).
    Zabai (D348, L: 4°3/4) = Singapore (real L: 1°.3).

  • The GD's failure to notice prominent Hainan Island (which nearly blocks off the east side of the broad Tonkin Gulf) suggests that the report Marinos used did not extend beyond Saigon (which is in fact the farthest point of Alexandros' narrative), so Alexandros & thus the GD never reached Hanoi or Hong Kong.

    Brief Comments & Hypotheses on Several Subjects

  • Parts of the GD show familiarity with the Euphrates River. (E.g., GD 1.12.5, 5.20.1-3&6.) So: how could GD 5.20.6 refer to Babylon as merely being “on the river that goes through Babylonia”? This appears to be just an unconsidered quick-info-transplant from an uncited source — and yet another hint of patch-workery.

  • Berggren & Jones 2000 p.44 notes that from GD 5.13 on, the most trustworthy ms (X) bears no coordinate data. Since the dataless lands were acquired late (after 100 BC) if at all by the Roman Empire, one might wonder if this oddity reflects dependence of the GD's data (up to that point) upon early Greco-Roman lists, maps, or globes. Perhaps of Hipparchos' epoch.

  • Marinos' ekumene was overbroad: a 225°-wide known-world, 5/8 of a wrap. This was justly revised (GD 1.12-14) to a smaller and much more accurate half-wrap breadth of 180° (see GD 1.14.10), though Berggren & Jones 2000 n.53 (p.76) rightly note the over-roundness here: Ptolemy aimed to get 180° — “by hook or by crook”.
    Had Marinos-Ptolemy not implicitly trusted (Rawlins 1985 n.14) E-W stade-measures over eclipse-measures of longitude (contra priority promo-announced at GD 1.4) and thus altered all degree-longitudes by a constant Earth-size-shift factor (Rawlins 1985 p.264) when switching from 700 stades/degree to 500 stades/degree, then the known-world's GD breadth in degrees would have been quite close to the truth — as was Ptolemy's breadth in distance (error merely ordmag 10% high): 90000 stades = 9000 nmi from BlestIsles-W.Europe to Java-E.China-Vietnam.

  • Thus, strangely (since latitudes were much easier for the ancients to measure accurately), the Ptolemy ekumene's longitudinal stades-distance-across is not less trustworthy than his latitudinal stades-distance-across.

  • We had a similar surprise earlier in finding longitude error-noise not worse than that in latitude. The upshot of both findings is an important broad insight:
    the merits of the GD are more geographical than astrographical.

  • Some scholars aver that an ambiguous discussion at Strabo 2.1.34-35 shows that Hipparchos knew Babylon's true latitude, 32°1/2. But the argument is vitiated by the high sensitivity of its key triangles' north-south sides, to slight uncertainties of ordmag 100 stades in other sides. Confirmatorily lethal: Strabo's very next paragraph (ibid 2.1.36) unambiguously, unsensitively reports that Hipparchos placed Babylon over 2500 stades north of Pelusium (D150), which was well-known (in reality [31°01'N] & at GD 4.5.11 [31°1/4]) to be near the same 31° parallel as Alexandria (GD 4.5.9). (Opposite sides of the Nile Delta: Alexandria-Canopus on the west, Pelusium on the east. Contiguous entries in GD 8.15: items 10&11 = D149&150, respectively.) At Hipparchos' 700 /1° scale, this puts Babylon (D256) rather north of 31°1/4 + 2500 stades/(700 stades/1°) = 34°5/6-plus — i.e., at 35°N, just the grossly erroneous value we find at GD 5.20.6 and (effectively) at GD8.20.27 — and on all other extant ancient Greek Important-City lists.
    [A consideration which alone could serve to gut the entire long-orthodox Neugebauer-group fantasy that high or even low Greek mathematical astronomy was derived from Babylon. Note that the same Strabo passage shows that Eratosthenes' latitude for Babylon was as erroneous as Hipparchos' but in the other direction. I.e., the entire Greek tradition had no accurate idea of where Babylon was, despite by-then long-standing contacts that had transmitted, e.g., invaluable Babylonian eclipse records.]
    More important to our present investigation: this finding leaves still-uncontradicted DR's proposal (Rawlins 1985 p.261) that Hipparchos was the ultimate source of the corrupt state of the GD's network's key latitudes.
    [It has been remarked that the Strabo 2.5.34 intro to his discussion of Hipparchos' klimata appears to state that Hipparchos was computing celestial phenomena every 700 stades (i.e., every degree) north of the Equator. But since the lengthy klimata data immediately following are instead almost entirely spaced at quarter-hour and half-hour intervals, DR presumes that the original (of the material Strabo was digesting) said that Hipparchos was providing latitudes (for each klima) in stades according to a scale of 700 stades/degree — a key attestation that Hipparchos had adopted Eratosthenes' scale.]

    Ptolemy's 1st Planar World-Map Projection:
    From Where-in-the-World Arrived
    That 34-Unit Vertical Strut (ε-to-η)
    from Its Top to Its “North Pole”?
    Ancient Averaging. And Weights?

        Ptolemy's 1st Projection. (Nobbe 1:47 [1843], ekumene's bound added in green.) Proceeding south, we successively encounter arcs representing the ekumene portions of six latitudinal circles: Thule = ξ-o-π, Rhodos = θ-κ-λ, Tropic of Cancer, Meroë, Equator = ρ-σ-τ, anti-Meroë = μ-ζ-ν.

  • In GD 1.24, Ptolemy twice attempts to design a planar portrayal of a broad spherical geographical segment, representing the known world — the ekumene — covering 180° of longitude from the Blest Isles (0° longitude) to easternmost China-Vietnam (180°E. longitude) and 79°5/12 (GD 1.10.1) of latitude from Thule [Shetlands (Mainland)] (63°N. latitude) to anti-Meroë (16°5/12 S. latitude, a parallel as far south of the Equator as Meroë's is north of the Equator).
    [Ptolemy rightly scaled-down Marinos' eastern limit from c.225° (15h = 5/8 of circle) to 180° (12h = 1/2 of circle); southern limit, from c.24° (Tropic of Capricorn) to 16°5/12 (anti-Meroë).]
    It is the 1st of his two projections (GD 1.24.1-9) which will concern us, since it involves a hitherto-unsolved mystery. This projection is a fan, opened slightly more than a right angle, c.98°. Thus, all north-latitude ekumene semi-circles are represented by 98° arcs. The fan is fairly neatly placed within a rectangle about twice as wide as high, as we see from the above illustration, where the four corners of the rectangle are (clockwise from upper left) points α, β, δ, γ.

  • For the 1st Projection's conversion of the spherical-segment ekumene to planarity, the degree-distance T = 63° from Equator to Thule is made into T = 63 linear units; likewise for the S = 16°5/12 from Equator to anti-Meroë, etc. In the projection (see illustration above), portions of several latitude-circles are drawn-in:
    the Thule circle (latitude 63°N) = ξ-o-π;
    the Rhodos circle (latitude 36°N) = θ-κ-λ;
    the Equator (latitude 0°) = ρ-σ-τ;
    the anti-Meroë circle (latitude 16°5/12 S) = μ-ζ-ν.

  • Beyond the Equator, instead of continuing to extend the radiating meridians of his fan-projection, Ptolemy decides to bend all meridians inward — resulting in the green-bounded ekumene we see in the above projection. This kink-step enables Ptolemy to force (GD 1.24.7) the length of the anti-Meroë parallel (south of the equator: latitude −16°5/12) to be exactly as long as its northern equivalent, the Meroë parallel (latitude +16°5/12).
    [This renders all the southern parallels of the GD ekumene virtually equivalent (in length, though not radius) to their northern counterparts.]

  • Ptolemy's angular↔linear duality here is effected by two rough expedients:
    [a] Defining the fan's units by forcing the distance T from Equator to Thule circle — 63 degrees of latitude — to be 63 units.
    (T = 63 is henceforth both a distance and an angle-in-degrees.)
    [b] Making the distance H, from the Thule circle to the fan's pseudo-N.Pole (point η in projection) proportional to cos 63° — i.e., equal to cos 63° in units of R, the fan's radius from “N.Pole” (point η) to Equator. (Simply put: H/R = cos 63°.)

    These conditions produce:

    T = RH = RRcosT = R(1 − cosT)


    R = T/(1 − cosT) = 115.38… ≈ 115.

    (The rounding is Ptolemy's.) Which produces the radius H of the Thule latitude-circle (centered at the pseudo-N.Pole η):

    H = RT = 115 − 63 = 52.

    Letting S = the south latitude of anti-Meroë, we further define

    E = R + S = 115 + 16 5/12 = 131 5/12

    This establishes all the fan's dimensions.
    [A list for ready reference. If we go up the mid-vertical of the projection, we find:
    o-η is of length H = 52 (as is ξ-η);
    σ-o is of length T = 63 (as is ρ-ξ);
    σ-η is of length R = 115 (as is ρ-η);
    ζ-σ is of length S = 16 5/12 (as is μ-ρ);
    ζ-η is of length E = 131 5/12 (as is μ-η);
    we recall that ε-η is of length Y;
    ζ-ε is of length Z, as are the sides of the rectangle: γ-α & δ-β.]

  • We next turn to the more puzzling question of how wide-open the fan will be.

  • The openness of the fan is immediately determined when Ptolemy states (GD 1.24.2) that he will choose a vertical strut Y = 34 units, extending from the top of the rectangle (bounding the fan) to the pseudo-N.Pole, which is the fan's radiating center. (I.e., the strut extends from the projection's point ε to its pseudo-N.Pole, point η.) And then — a very strange step appears.

  • Since Ptolemy follows Hipparchos and (GD 1.20.5) Marinos in taking the Rhodos latitude (36°) or klima (14h1/2) as canonical for the mid-ekumene, he chooses (GD 1.24.3;) the Rhodos parallel at latitude 36°N as the one along which he will (allegedly) adjust longitudinal distances precisely, just so that this parallel's curved length (west-east arc) has the correct proportion (4:5 ≈ cos 36°: GD 1.20.5 & 24.3) to the fan's already-determined north-south radial distances.

  • That step is odd because, when he earlier established Y = 34 units, this rigidly fixed the fan's openness, and thus the proportion along the Rhodos parallel. I.e., there is no fan-openness flexibility left, once Y is set at 34 units.

  • Well, you may suppose: Ptolemy must have chosen Y = 34 with this very point in mind — this of course has to be the precise value for Y which will ensure proper Rhodos-parallel proportionality. But, no. He didn't, and it isn't. We can tell so by just doing the math.

  • If we let L be the latitude of Rhodos or any other place, the following equation finds that value of Y which will guarantee the desired proportionality at the given L's parallel:

    Y = Hcos{16200cos L/[π(R − |L|)]}

    (L's sign-insensitivity in this equation is due to Ptolemy's kink-step.)

  • But the truth swiftly reveals itself when we substitute Rhodos' L (36°) into this equation: we get Y ≈ 31 units (nearly 32 without Ptolemy's rounding) — not 34 units.
    [Y = 31 corresponds to fan-spread 106° not 98°.
    (F = Fan-spread = 2arccos(Y/H) = 32400cos L/[π(R - L)].) This accounts for the non-fitting & unintended aggravation that points ξ&π lie above the top (α&β) of the rectangle in several modern depictions of the situation. (The discrepancy has long been recognized; see, e.g., Wilberg & Grashof p.78.) The screwup is not by the drafters but by Ptolemy, who did not realize that 34 units is not for the Rhodos parallel (corresponding to the 106° fan-spread used by the non-fitting diagrams just cited) but was designed as an average fit to all ekumene parallels. Note that for L = 0° (Equator) or 63° (Thule), fan-spread would be 90° (Y ≈ 37). The average of 106° & 90° is 98°, which fits Y = 34 (the average of 31&37).]

  • Two obvious questions now arise:
    [a] Why didn't Ptolemy know the origin of Y = 34?
    [b] What, then, is the true origin of his choice of Y = 34?

  • The answers are:
    [a] Because as usual Ptolemy plagiarized math he didn't understand the origin of.
    [b] We get a clue to the actual origin when we substitute other latitudes L into the foregoing equation: we find that Y reaches a minimum very near Rhodos — and is considerably higher near the Tropics or the Arctic. The Y for Thule (L = T = 63°) is the same as for the Equator (L = 0°), since substituting either of these two L-values into the general equation reduces it to:

    Y = Hcos[16200/(πR)] ≈ 37

  • Noting that the mean of our last two results is (31 + 37)/2 = 34, we may now commence our upcoming solution-reconstruction of the insights of the actual designer of the fan-map Ptolemy swiped.

  • The 1st thing the true originator presumably noticed was that, in order to arrive at a meaningful averaged Y-value, it made no sense to use (as Ptolemy claims to) a mid-ekumene parallel — since the solutions for Y did not increase linearly or even monotonically in the latitude-range under consideration. Instead, if we go south: the values for the Y that are apt (i.e., produce correct longitudinal proportion) start at Y ≈ 37 for Thule, dip to a minimum of about 31 almost exactly at Rhodos, and then double right back up to 37 for the Equator. So the obvious crude solution was to average 31 and 37, yielding 34.

  • Better: a mean Y for all ekumene latitudes also = 34. With or without rounding. If we go on to a truly proper solution and use weightings by area (since tropical latitude-intervals contain more area than non-tropical), we still find that mean Y ≈ 34. (Again: with or without rounding.)
    [If we eliminate the southern latitudes, we yet find Y ≈ 34, except for the non-weighted average with rounding, where Y ≈ 33 1/3 instead.]
    I.e., the result is a firm one, encouraging the hypothesis that we have here successfully induced the true origin of Ptolemy's strut-length: Y = 34, an origin of which he was evidently unaware. Moreover, the result is consistent with (though it does not prove) ancient mathematical mappers' attention to proportional preservation of areas (even if but imperfectly), a consideration for which no evidence has previously been in hand.
    [See, e.g., Berggren & Jones 2000 p.38.]

    The Impossible Dream of the Symmetric-Rectangle-Bounded Ekumene Fan

  • There is an attractive alternate theory of the origin of Y = 34: the suggestion that the 2-1 rectangle bounding Ptolemy's ekumene influenced the openness of the fan: “The length of 34 units … seems to have been empirically chosen to accommodate the largest map in the given [2-1] rectangle without truncation of the corners [ρ&τ].” (Berggren & Jones 2000 p.86 n.68.) We will now explore this theory, which takes us in a very different (but equally fascinating) direction from the previous section.

  • Ptolemy says his projection nearly fits neatly into a 2-1 landscape-oriented rectangle: see above illustration (Nobbe 1:47 [1843]).
    Since the fan-projection is symmetric about the mid-vertical (ε-ζ), the rectangular condition can be equated with fitting the left or right half of the ekumene into a split-off square. (Splitting the rectangle into halves, we will use the left square during the following analysis). Fitting the half-ekumene into a square will henceforth be referred to here as: the split-constraint or just The Split.

  • Having arranged that each half of the projection's rectangular bound is a perfect square of side Z, we take half of the horizontal straight line between ρ & τ and call it B. Note: if The Split-condition is met, then B should equal half of the rectangle's top border (α-β). But it obviously does not, for reasons to be seen.
    Our aim is to (as closely as possible) meet the Split-condition, which can be expressed simply as:

    Z = B

    [Notice to those checking-via-ruler the (Nobbe) projection's rectangle: its halves are accidentally drawn not quite square, though very close.]

  • We then search for the value of Y which ensures that Ptolemy's ekumene-fan will satisfy The Split. The equation is (using the inputs already defined):

    Y = [E + (R/H)(R2 + H2E2)1/2]/[(R/H)2 + 1]

  • Ptolemy starts by assuming that the meridian-radiating center of the fan (the pseudo-N.pole: point η in the above projection) is Y = 34 units (GD 1.24.2) above the top of the rectangle that he proposes to contain his ekumene projection. (To repeat, we are saying that in the above projection, the distance from η to ε = 34 units.)

  • Ptolemy admits (GD 1.24.1) that his 2-1 rectangle isn't quite exact: the rectangle's width is only nearly (εγγιστα) two-fold its width.
    [For the 2nd projection, there is no such qualifier (GD 1.24.17), even though there might as well have been — since for both projections the 2-1 rectangular bound is slightly wider than necessary. But for the 2nd projection, there is no appearance that an adjustment might render the ekumene exactly twice as wide as high. Its definition is quite different from the 1st, and results in a fan opened only about 61° (vs the 1st's 98°), with a pseudo-north-pole c.180 units (vs the 1st's 115 units) above the Equator.]
    But: why only approximately twice as wide? Why not adjust Y such as to make the ratio exact? — since the priority here is suspected to be The Split: a symmetric 2-1 rectangularly-bounded fan, for reasons either aesthetic (symmetry) or practical. (A portable map that is conveniently square after one protective fold?)

  • The hitherto-unrecognized answer is that, given Ptolemy's specs for the projection's essentials (T = 63 and S = 16 5/12), the 2-1 rectangle-bound condition for the fan cannot be met. Mathematically speaking: for the cited Ptolemaic values of T&S, the only solutions for Y that can result from the above equation are not real. This surprise finding will now lead us onto unexpected paths.

  • I.e., the ekumene-fan as Ptolemy ultimately constructed it cannot fit into a 2-1 rectangle, no matter how widely or narrowly the Thule-bounded ekumene-fan is fanned out, so long as S = 16 5/12. Try it for yourself. As S is increased, we find (from the above equation) that the maximum ekumene southern-limit S that allows Y = 34 and satisfies the symmetry of The Split is about S = 6.

  • When the Fan Fit The Split:
    So the 2-1 theory has exploded in disaster: no choice of Y will satisfy Ptolemy's S = 16 5/12 and allow the fan-projection to fit the symmetric 2-1 rectangle. Indeed, the maximum S that will permit satisfaction of The Split (for any choice of Y) is found via the equation:

    Smax = T[(1 + cos2T)1/2 − 1]/(1 − cosT)

    which for T = 63 (fan's north bound at Thule) yields Smax ≈ 11 1/3.

  • Things get even more intriguing if we assume (as some non-adamantly have) that Y = 34 was an empirical adjustment to The Split (the 2-1 rectangle condition). We can test the theory by finding the value of Y which best satisfies The Split.
    Answer: Y ≈ 21 — a value not even close to 34.
    Y ≈ 21 satisfies The Split to within 5%: that is, Z/B < 1.05. But Ptolemy's Y = 34 cannot satisfy the 2-1 rectangle condition to better than 11%, i.e., Z/B > 1.11.

  • However, let's keep exploring the theory that the 34 was chosen for The Split. (If Ptolemy was seeking any other type of symmetry, the obvious and nearby alternative would have been to make the fan-spread angle [ξ-η-π] equal to exactly 90° — not the seemingly pointless and peculiar [roughly 98°] spread we actually find: see projection.) A 90° spread would make all longitude slices neatly 1/2 their real angular thickness.
    [The corresponding Y = H/√2 = 37, obviously not Ptolemy's choice.]

  • Our math for an attempted Split-inspired reconstruction of the process behind Y = 34 will, up to a point, be the same as Ptolemy's — only simpler.
    We round R = 115.4 to 115 (just as at GD 1.24.4) but then use a simple fan — i.e., without Ptolemy's equatorial kink.
    [That is, we do not immediately follow Ptolemy in suddenly bending all meridians inward after southward-crossing the Equator. That step eliminated (for Ptolemy) the extreme-outside points μ&ν. But we instead keep it simple by letting lines η-ρ and η-τ extend right straight out to μ and ν, respectively — and leave them be (i.e., no kink) — just as these two points are shown (slightly outside the 2-1 rectangle) in the projection.]

  • Once we dispense with Ptolemy's clever kinky-projection scheme, we may easily find the S that produces Y = 34:

    S = H2{1 + [1 − (Y/H)2]1/2}/Y − R

    Substituting Ptolemy's values, Y = 34 (GD 1.24.2) and R = 115 & H = 52 (GD 1.24.4), we find:

    S = 24.7

    A provocative result, since that is virtually right on the southern tropic (24°).

  • However, as noted: S = 24° is Marinos' value — according to Ptolemy himself (GD 1.7.1-2 & 9.6). Thus, we have found a potentially fruitful alterate-possibility for the source of the problematic Y = 34: a non-kinked fan-ekumene, with Marinos' latitudinal breadth of the known world, though Marinos is said (GD 1.20.4-5) not to have used a fan-projection.

  • Having thus found an S that could have led to GD 1.24.2's Y = 34, we may simply invert the process to follow in the hypothetical math-footsteps of the hypothetical ancient scholar who hypothetically deduced said Y. If we also dispense with intermediate variables, to show dependence purely upon the ekumene's northern & southern limits (T & S, resp), the inverse of the previous equation gives us what we need:

    Y = 2[S + T/(1 − cosT)]/{1 + [(1 + S{1 − cosT}/T)/cosT]2}

    Substituting (into the above equation) T = 63 (Thule) and S = 24 (southern tropic), the hypothetical ancient computer (of the Y that has come through to us) found

    Y = 34

    (Barely less than 34 1/2 without Ptolemy's rounding of R to 115; or about 34 1/8, if that rounding is adopted.)

  • But GD 1.24.4-5 denies that Marinos used the fan-scheme. If this report is to be trusted and if the Split-hypothesis is valid, then: at an early stage in the history of the development of the fan-approach, a scholar (working sometime between Marinos and the final version of GD 1.24) tried out a simple (no-kink) fan using Marinos' southern limit (S = 24).

  • However, had he adopted S = 16 5/12 without kinking his projection, he could easily have found (using the previous equation) that for this case the appropriate Y = 36, which would in fact effect a perfect-Split circumscription of the (non-kinked) fan by the preferred symmetric 2-1 rectangle.
    [Berggren & Jones 2000 p.87 n.69 point out the oddity that the GD 1.24 discussion refers only to pt.υ not pt.ζ, though they are identical. (Both are shown on the projection.) This would appear to indicate that at some drafting point, before arrival at the final version of the first projection, pts.υ&ζ were separate. This could have happened during experiments ere the kink (when the 2-1 rectangle touched pts.μ&ν) or ones where the projection's southern parallel was the Equator or the Tropic of Capricorn.]

  • So, if the Split-theory is valid, Y must have been frozen at 34 before any steps were taken to abandon either
    [1] assumption of S = 24 (Marinos), or
    [2] the simple non-kinked fan-scheme.

  • If Ptolemy adopted Y = 16 5/12 before kinking his fan, then he could easily have arrived at Y = 36 by the same means that 34 was arrived at. (As already shown above.) Since 36 is not what survived, it would follow that Ptolemy instead kinked his fan before bringing his southern boundary from Y = 24 up to 16 5/12.

  • However, either way, he at some point would be faced with the problem of finding out what Y would most closely effect The Split if the kinked version of his ekumene projection were adopted. For this search, he had best be aware that the Split ratio (Z/B) is maximal when (on the projection) a line drawn from ζ to ξ is perpendicular to the radial line η-μ. Thus, the best fit to The Split occurs when:

    Y = H2/E

    For S = 16 5/12, this equation yields, as noted previously, Y ≈ 21, which corresponds to fan-spread F = 132°. For S = 24, Y ≈ 20 — corresponding to F = 135°.

  • Even if the foregoing Split-theory isn't historical (and the previous section — much-preferred by DR — obviously assumes that it is not), the mathematical development of it here has been thoroughly enjoyable.

    The entire 3-volume Nobbe text is available in pdf form at Just fill “Nobbe” in the Search-box there. On the 1st page that appears, the several consecutive links for “Claudii Ptolemaei Geographia Page One” are simply for various sections of the Nobbe work.
    vol.1 (1843) = Books 1-4;
    vol.2 (1845) = Books 5-8 plus collation to A.Montani ed.'s pagination;
    vol.3 (1845) = 3 indexes.
    Book 8 begins at vol.2 p.191.

    The sections on Inter-Relations and on Precession were first internet-posted 2006/11/18. Section on China, 2006/11/29.
    The other sections were 1st posted 2007/4/30.

    [Intermittent revisions of all: 2006/11/30-2008/11/12.]