Although the fact that the Earth is a sphere was already known by Greek philosophers, astronomers and geographers, the concept of a calendrical date line or the fact that a traveller would either gain or lose a day after circumnavigating the globe does not seem to surface until the early 12th century.
What appears to be the earliest reference to the concept of a calendrical date line is found in a treatise first written around 1140 in Arabic by the Spanish-Jewish philosopher and poet Rabbi Yehuda ben Shemuel Ha-Levi (c. 1075-1141). In the second chapter (par. 18-20) of his Kitab al-Hujja waal-Dahl fi Nasr al-Din al-Dhalil (The Book of Argument and Proof in Defence of the Despised Faith), better known from the title of its translation in 1167 by Rabbi Judah ben Saul ibn Tibbon (1120-after 1190) into Hebrew as the Sefer ha-Kuzari, Ha-Levi discusses the halaka (Jewish Law) concerning the beginning of the Jewish Sabbath.
In order that the Jews in the diaspora would observe the Sabbath on the same day as the Jews in Israel, Ha-Levi argued that although according to Jewish tradition the day begins [at sunset] in Israel (i.e. the Jewish Date Line runs through Israel), all the regions to the east of Israel up to the eastern shores of the Asian continent should, despite their difference in longitude, be considered to share the same time and day reckoning.
The earliest reference to the so-called circumnavigators paradox is found in a treatise on geography by the Syrian prince and geographer-historian Ismail ibn Ali ibn Mahmud ibn Muhammad ibn Taqi ad-Din Umar ibn Shahanshah ibn Ayyub al Malik al Muayyad Imad ad-Din Abu l-Fida (1273-1331). In the introductory chapter to his Taqwīm al-Buldān (Geography), Abu l-Fida described how a traveller, depending on the direction of travel, would either lose or gain a day at the completion of his circumnavigation.
Let us assume that it is possible to make a journey around the world. Suppose further that three persons meet in a fixed place one then sets off to the west, another sets off to the east while the third remains where he is and waits for the other two to complete their journey around the world. The one who set off to the west will return from the east while the one who set off to the east will return from the west. But the one who travelled to the west will have lost a day while the one who travelled to the east will count a day too many.
Indeed, the one who travelled to the west (we will assume it takes seven days to travel around the earth) walked in the same direction as the Sun, so for him the Sun set a seventh part of the day later each day. This, at the end of seven days, made a full revolution, or a full day.
The one who travelled to the east pursued a direction opposite to that of the Sun, so for him the Sun set a seventh part of the day earlier each day. This, at the end of seven days, made up a complete day which obliged him to count an additional day.
Thus, if the day of departure was a Friday and the day when the travellers met again was the following Friday, as counted by the person who had remained stationary, the one who had travelled to the west and returned from the east will reckon it to be a Thursday, while the one who had travelled to the east and returned from the west will reckon it to be a Saturday. The result will be the same, if instead of several days, the journey around the world would had lasted months or even years.
Further early references to the circumnavigators paradox are found in several works of the influential French philosopher Nicole Oresme (c. 1323-1382). In his Traitié de lespere (which was also translated into Latin as the Tractatus sperae), Oresme presented a remarkable circling of the Earth by two imaginary travellers Jehan and Pierre (Johannes and Petrus in the Latin version) who set out to journey around the world along the equator in opposite directions at a speed of 30 degrees of terrestrial longitude per 24-hour day. Jehan, travelling in a westward direction, would claim at the completion of his journey that it took him only eleven days and nights while Pierre, travelling in an eastward direction, maintained that his journey lasted thirteen days and nights. A third man, Robert, who had remained at the starting point, however would point out that only twelve days and nights had elapsed since both travellers had set out.
Oresme repeated this argument in his Quaestiones supra speram, a series of clarifications of questions based on the popular cosmographical treatise Tractatus de Sphaera by Johannes de Sacrobosco (c. 1195-c. 1256), in which he renamed his travellers Plato and Socrates and the control Petrus and allowed both travellers a more leisurely pace of 14.4 degrees of terrestrial longitude per 24-hour day. At the return of the philosophers at the starting point, Plato (the westward traveller) would have logged twenty-four days, Socrates (the eastward traveller) no less than twenty-six days, while Petrus had seen the sun rise and set only twenty-five times.
In order to resolve the circumnavigators paradox for future travellers, Oresme concluded his discussion of the imaginary journeys of Plato and Socrates in the Quaestiones supra speram with the observation:
From this it follows that if this [equatorial] zone were everywhere habitable, one ought to assign a definite place where a change of the name of the day would be made, for otherwise Socrates would have two names for the same day and the other [Plato] would have the same name for two days.
Around 1377 Oresme wrote his Traitié du ciel et du monde, a French translation and commentary of Aristotles De caelo et mundo, in which he again discussed the circumnavigators paradox. Here the westward traveller is simply named A, the eastward traveller B and the control C. Each of both travellers is now assumed to cover 40 degrees of terrestrial longitude per 24-hour day; A counting eight days for his circumnavigation, B ten days, while C only marks nine days on his calendar.