Editing Pytheas (section)

== Measurements of latitude ==

=== Latitude by the altitude of the Sun ===
In discussing the work of Pytheas, Strabo typically used direct discourse: "Pytheas says&nbsp;..." In presenting his astronomical observations, he changed to indirect discourse: "[[Hipparchus]] said that Pytheas says&nbsp;..." either because he never read Pytheas' manuscript (because it was not available to him) or in deference to Hipparchus, who appears to have been the first to apply the Babylonian system of representing the sphere of the Earth by 360°.<ref>{{cite book|title=Surveying Instruments of Greece and Rome|first=Michael Jonathan Taunton|last=Lewis|publisher=Cambridge University Press|date=2001|location=Cambridge, New York|isbn=978-0-521-79297-4|pages=26–27}}</ref>

Strabo used the degrees, based on Hipparchus.<ref>''Geographica'' [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2E2*.html II.5.34]: "If, then, we cut the greatest circle of the Earth into three hundred and sixty sections, each of these sections will have seven hundred stadia."</ref> Neither say that Pytheas did. Nevertheless, Pytheas did obtain latitudes, which, according to Strabo, he expressed in proportions of the gnōmōn ("index"), or trigonometric [[Trigonometric functions|tangents]] of angles of elevation to celestial bodies. They were measured on the gnōmōn, the vertical leg of a right triangle, and the flat leg of the triangle. The imaginary hypotenuse looked along the line of sight to the celestial body or marked the edge of a shadow cast by the vertical leg on the horizontal leg.

Pytheas took the altitude of the Sun at Massalia at noon on the longest day of the year and found that the tangent was the proportion of 120 (the length of the gnōmōn) to 1/5 less than 42 (the length of the shadow).<ref>''Geographica'' [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2E2*.html II.5.41].</ref> Hipparchus, relying on the authority of Pytheas (says Strabo<ref>II.1.12 and again in II.5.8.</ref>), states that the ratio is the same as for [[Byzantium]] and that the two therefore are on the same parallel. Nansen and others prefer to give the [[cotangent]] 209/600,<ref name=nansen46>{{harvnb|Nansen|1911|p=46}}.</ref> which is the inverse of the tangent, but the angle is greater than 45° and it is the tangent that Strabo states. His number system did not permit him to express it as a decimal but the tangent is about&nbsp;2.87.

It is unlikely that any of the geographers could compute the [[arctangent]], or angle of that tangent. Moderns look it up in a table. Hipparchos is said to have had a table of some angles. The altitude, or angle of elevation, is 70°&nbsp;47′&nbsp;50″<ref name=nansen46 /> but that is not the [[latitude]].

At noon on the longest day the plane of longitude passing through Marseille is exactly on edge to the Sun. If the Earth's axis were not tilted toward the Sun, a vertical rod at the [[equator]] would have no shadow. A&nbsp;rod further north would have a north–south shadow, and as an elevation of 90° would be a zero latitude, the [[Complementary angles|complement]] of the elevation gives the latitude.{{Citation needed|reason=Either explain better or cite someone who does.|date=December 2023}} The Sun is even higher in the sky due to the tilt. The angle added to the elevation by the tilt is known as the [[Axial tilt#Obliquity|obliquity of the ecliptic]] and at that time was 23°&nbsp;44′&nbsp;40″.<ref name=nansen46 /> The [[complementary angles|complement]] of the elevation less the obliquity is 43°&nbsp;13′, only 5′ in error from Marseille's latitude, 43°&nbsp;18′.<ref>Most students of Pytheas presume that his differences from modern calculations represent error due to primitive instrumentation. Rawlins assumes the opposite, that Pytheas observed the sun correctly, but his observatory was a few miles south of west-facing Marseille. Working backward from the discrepancy, he arrives at Maire Island or Cape Croisette, which Pytheas would have selected for better viewing over the south horizon. To date there is no archaeological or other evidence to support the presence of such an observatory; however, the deficit of antiquities does not prove non-existence. {{cite journal|title=Pytheas' Solstice Observation Locates Him|first=Dennis|last=Rawlins | journal=DIO & the Journal for Hysterical Astronomy | volume=16 | date=December 2009 | pages=11–17 | url=http://www.dioi.org/vols/wg0.pdf}}</ref>

=== Latitude by the elevation of the north pole ===
A second method of determining the latitude of the observer measures the angle of elevation of a [[celestial pole]], north in the northern hemisphere. Seen from zero latitude the north pole's elevation is zero; that is, it is a point on the horizon. The [[declination]] of the observer's [[zenith]] also is zero and therefore so is their latitude.

As the observer's latitude increases (traveling north) so does the declination. The pole rises over the horizon by an angle of the same amount. The elevation at the terrestrial [[North Pole]] is 90° (straight up) and the celestial pole has a declination of the same value. The latitude also is 90.<ref>{{cite book|title=The American Practical Navigator: an Epitome of Navigation|first=Nathaniel|last=Bowditch|author-link=Nathaniel Bowditch|publisher=National Geospatial-Intelligence Agency |date=2002 | edition=Bicentennial  |url=http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/APN/Chapt-15.pdf |page=243 |quote=That is, the altitude of the elevated pole is equal to the declination of the zenith, which is equal to the latitude|access-date=7 June 2012}}</ref>

Moderns have [[Polaris]] to mark the approximate location of the North celestial pole, which it does nearly exactly. This was not the case in Pytheas' time, due to the [[precession of the equinoxes]]. Pytheas reported that the pole was an empty space at the corner of a quadrangle, the other three sides of which were marked by stars.<ref>The report survives in the ''Commentary on the Phainomena of Aratos and Eudoxos'', 1.4.1, fragments of which are preserved in [[Hipparchos]].</ref> Their identity has not survived but based on calculations these are believed to have been α and κ in [[Draco (constellation)|Draco]] and β in [[Ursa Minor]].<ref>{{cite web | first=T.E. | last=Rihll | title=Greek and Roman Science and Technology, V3; Specific subjects; Astronomy | url=http://www.swan.ac.uk/grst/Home%20Page%20G&RS&T.htm | publisher=Swansea University | access-date=7 June 2012 | location=Note 14 | archive-date=6 July 2012 | archive-url=https://web.archive.org/web/20120706153833/http://www.swan.ac.uk/grst/Home%20Page%20G%26RS%26T.htm | url-status=dead }}</ref>

Pytheas sailed northward with the intent of locating the Arctic Circle and exploring the "frigid zone" to the north of it at the extreme of the Earth. He did not know the latitude of the circle in degrees. All he had to go by was the definition of the frigid zone as the latitudes north of the line where the celestial arctic circle was equal to the celestial Tropic of Cancer, the ''tropikos kuklos'' (refer to the next subsection). Strabo's angular report of this line as being at 24° may well be based on a tangent known to Pytheas, but he did not say that. In whatever mathematical form Pytheas knew the location, he could only have determined when he was there by taking periodic readings of the elevation of the pole (''eksarma tou polou'' in Strabo and others).{{citation needed|date=December 2017}}

Today the elevation can be obtained easily on ship with a [[Quadrant (instrument)|quadrant]]. Electronic navigational systems have made even this simple measure unnecessary. Longitude was beyond Pytheas and his peers, but it was not of as great a consequence, because ships seldom strayed out of sight of land. East–west distance was a matter of contention to the geographers; they are one of Strabo's most frequent topics. Because of the [[Gnomon|gnōmōn]] north–south distances were accurate often to within a degree.{{citation needed|date=December 2017}}

It is unlikely that any gnōmōn could be read accurately on the pitching deck of a small vessel at night. Pytheas must have made frequent overnight stops to use his gnōmōn and talk to the natives, which would have required interpreters, probably acquired along the way. The few fragments that have survived indicate that this material was a significant part of the ''periplus'', possibly kept as the ship's log. There is little hint of native hostility; the Celts and the Germans appear to have helped him, which suggests that the expedition was put forward as purely scientific. In any case all voyages required stops for food, water and repairs; the treatment of voyagers fell under the special "guest" ethic for visitors.{{citation needed|date=December 2017}}

=== Location of the Arctic Circle ===
The ancient Greek view of the heavenly bodies on which their navigation was based was imported from [[Babylonia]] by the [[Ionia]]n Greeks, who used it to become a seafaring nation of merchants and colonists during the [[Archaic period in Greece]]. Massalia was an Ionian colony. The first Ionian philosopher, [[Thales]], was known for his ability to measure the distance of a ship at sea from a cliff by the very method Pytheas used to determine the latitude of Massalia, the trigonometric ratios.

The astronomic model on which ancient Greek navigation was based, which is still in place today, was already extant in the time of Pytheas, the concept of the degrees only being missing. The model<ref>''Geographica'' [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2E1*.html II.5.3].</ref> divided the universe into a celestial and an earthly sphere pierced by the same poles. Each of the spheres were divided into zones (''zonai'') by circles (''kukloi'') in planes at right angles to the poles. The zones of the [[celestial sphere]] repeated on a larger scale those of the terrestrial sphere.

The basis for division into zones was the two distinct paths of the heavenly bodies: that of the stars and that of the Sun and Moon. Astronomers know today that the Earth revolving around the Sun is tilted on its axis, bringing each hemisphere now closer to the Sun, now further away. The Greeks had the opposite model, that the stars and the Sun rotated around the Earth. The stars moved in fixed circles around the poles. The Sun moved at an oblique angle to the circles, which obliquity brought it now to the north, now to the south. The circle of the Sun was the [[ecliptic]]. It was the center of a band called the [[zodiac]] on which various constellations were located.

The shadow cast by a vertical rod at noon was the basis for defining zonation. The intersection of the northernmost or southernmost points of the ecliptic defined the axial circles passing through those points as the two tropics (''tropikoi kukloi'', "circles at the turning points") later named for the zodiacal constellations found there, [[Tropic of cancer|Cancer]] and [[Tropic of Capricorn|Capricorn]]. During noon of the [[summer solstice]] (therinē tropē) rods there cast no shadow.<ref>''Geographica'' [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2E1*.html II.5.7].</ref> The latitudes between the tropics were called the torrid zone (''diakekaumenē'', "burned up").

Based on their experience of the Torrid Zone south of [[Egypt]] and [[Libya]], the Greek geographers judged it uninhabitable. Symmetry requires that there be an uninhabitable Frigid Zone (''katepsugmenē'', "frozen") to the north and reports from there since the time of [[Homer]] seemed to confirm it. The edge of the Frigid Zone ought to be as far south from the [[North Pole]] in latitude as the Summer Tropic is from the [[Equator]]. Strabo gives it as 24°, which may be based on a previous tangent of Pytheas, but he does not say. The Arctic Circle would then be at 66°, accurate to within a degree.<ref>Strabo's extensive presentation of the geographic model including the theory of the Arctic is to be found in [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2E1*.html Book II Chapter 5].</ref>

Seen from the equator the celestial North Pole (''boreios polos'') is a point on the horizon. As the observer moves northward the pole rises and the [[circumpolar stars]] appear, now unblocked by the Earth. At the Tropic of Cancer the radius of the circumpolar stars reaches 24°. The edge stands on the horizon. The [[constellation]] of ''mikra arktos'' ([[Ursa Minor]], "little bear") was entirely contained within the circumpolar region. The latitude was therefore called the ''arktikos kuklos'', "circle of the bear". The terrestrial Arctic Circle was regarded as fixed at this latitude. The celestial Arctic Circle was regarded as identical to the circumference of the circumpolar stars and therefore a variable.

When the observer is on the terrestrial Arctic Circle and the radius of the circumpolar stars is 66° the celestial Arctic Circle is identical to the celestial [[Tropic of Cancer]].<ref>{{harvnb|Nansen|1911|p=53}}.</ref> That is what Pytheas means when he says that Thule is located at the place where the Arctic Circle is identical to the [[Tropic of Cancer]].<ref name=straboII-5-8 /> At that point, on the day of the [[Summer Solstice]], the vertical rod of the [[Gnomon|gnōmōn]] casts a shadow extending in theory to the horizon over 360° as the Sun does not set. Under the pole the Arctic Circle is identical to the Equator and the Sun never sets but rises and falls on the horizon. The shadow of the gnōmōn winds perpetually around it.

=== Latitude by length of longest day, and by Sun's elevation on shortest day ===
Strabo used the astronomical cubit (''pēchus'', the length of the forearm from the elbow to the tip of the little finger) as a measure of the elevation of the Sun. The term "cubit" in this context is obscure; it has nothing to do with distance along either a straight line or an arc, does not apply to celestial distances, and has nothing to do with the gnōmōn. [[Hipparchus]] borrowed this term from [[Babylonia]], where it meant 2°. They in turn took it from ancient [[Sumer]] so long ago that if the connection between cubits and degrees was known in either Babylonia or Ionia it did not survive. Strabo stated degrees in either cubits or as a proportion of a [[great circle]]. The Greeks also used the length of day at the summer solstice as a measure of latitude. It is stated in equinoctial hours (''hōrai isēmerinai''), one being 1/12 of the time between sunrise and sunset on an [[equinox]].

Based partly on data taken from Pytheas, Hipparchus correlated cubits of the Sun's elevation at noon on the winter solstice, latitudes in hours of a day on the summer solstice, and distances between latitudes in stadia for some locations.<ref>{{harvnb|Nansen|1911|p=52}}.</ref> Pytheas had proved that Marseille and Byzantium were on the same parallel (see above). Hipparchus, through Strabo,<ref>Strabo [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2A1*.html II.1.12,13].</ref> added that Byzantium and the mouth of the Borysthenes, today's [[Dnieper]], were on the same [[Meridian (geography)|meridian]] and were separated by 3700 stadia, 5.3° at Strabo's 700&nbsp;stadia per a degree of [[meridian arc]]. As the [[Circle of latitude|parallel]] through the river-mouth also crossed the coast of "Celtica", the distance due north from Marseille to Celtica was 3700 stadia, a baseline from which Pytheas seems to have calculated latitude and distance.<ref>However, Srabo [https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/2A1*.html II.1.18] implied 3800, still attributed to Hipparchus. [[Eratosthenes]] has quite a different view. See under [[Thule]].</ref>

Strabo said that Ierne (written Ἰέρνη, meaning Ireland<ref>[[Éire#Etymology]]</ref>) is under 5000 stadia (7.1°) north of this line. These figures place Celtica around the mouth of the [[Loire]], an emporium for the trading of British tin. The part of Ireland referenced is the vicinity of [[Belfast]]. Pytheas then would either have crossed the [[Bay of Biscay]] from the coast of Spain to the mouth of the Loire, or reached it along the coast, crossed the [[English Channel]] from the vicinity of [[Brest, France]] to [[Cornwall]], and traversed the [[Irish Sea]] to reach the [[Orkney]] Islands. A statement of Eratosthenes attributed by Strabo to Pytheas, that the north of the [[Iberian Peninsula]] was an easier passage to Celtica than across the Ocean,<ref>[https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/3B*.html Strabo III.2.11].</ref> is somewhat ambiguous: apparently he knew or knew of both routes, but he does not say which he took.

At noon on the [[winter solstice]] the Sun stands at 9 cubits and the longest day on the [[summer solstice]] is 16 hours at the baseline through Celtica.<ref>Strabo II.1.18. The notes of the Loeb Strabo summarized and explained this information.</ref> At 2500&nbsp;stadia, approximately 283 miles, or 3.6°, north of Celtica, are a people Hipparchus called Celtic, but whom Strabo thought were the British, a discrepancy he might not have noted if he had known that the British were also Celtic. The location is Cornwall. The Sun stands at 6&nbsp;cubits and the longest day is 17&nbsp;hours. At 9100&nbsp;stadia, approximately 1032&nbsp;miles, north of Marseille, 5400 or 7.7° north of Celtica, the elevation is 4&nbsp;cubits and the longest day is 18&nbsp;hours. This location is in the vicinity of the [[Firth of Clyde]].

Here Strabo launched another quibble. Hipparchus, relying on Pytheas, according to Strabo, placed this area south of Britain, but he, Strabo, calculated that it was north of Ireland (Ierne/Ἰέρνη). Pytheas, however, rightly knew what is now [[Scotland]] as part of Britain, land of the [[Picts]], even though north of Ireland/Ierne. North of southern Scotland the longest day is 19 hours. Strabo, based on theory alone, states that Ierne is so cold<ref name=straboII-5-8 /> that any lands north of it must be uninhabited. In the hindsight given to moderns Pytheas, in relying on observation in the field, appears more scientific than Strabo, who discounted the findings of others merely because of their strangeness to him. The ultimate cause of his skepticism is simply that he did not believe Scandinavia could exist. This disbelief may also be the cause of alteration of Pytheas' data.