Pytheas : biography
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).Geographica . Hipparchus, relying on the authority of Pytheas (says StraboII.1.12 and again in II.5.8.), 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,. 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 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° 47’ 50″ but that is not the latitude.
At noon on the longest day the plane of longitude passing through Marseilles 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 rod further north would have a north-south shadow, and as an elevation of 90° would be a zero latitude, the complement of the elevation gives the latitude. 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 obliquity of the ecliptic and at that time was 23° 44′ 40″. The complement of the elevation less the obliquity is 43° 13′, only 5′ in error from Marseilles’s latitude, 43° 18′.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 Marseilles. 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.
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 his latitude.
As the observer’s latitude increases (he travels 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.
Moderns have Polaris to mark the approximate location of the North celestial pole, which it does nearly exactly, but this position of Polaris was not available in Pytheas’ time, due to changes in the positions of the stars. 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.The report survives in the Commentary on the Phainomena of Aratos and Eudoxos, 1.4.1, fragments of which are preserved in Hipparchos. Their identity has not survived but based on calculations these are believed to have been α and κ in Draco and β in Ursa Minor.
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 does 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).