Editing Pytheas (section)

=== 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>