Claudius Ptolemaeus; Giovanni Antonio Magini — Geographiae universae tum veteris, tum novae (1597)

chapter 109 of the second book on the heavens and the world records that a letter was found in the tomb of Dionysiodorus, which he wrote while alive, affirming that he had discovered the distance from the tomb to the lowest part of the Earth to be 42,000 stadia units of distance; according to this calculation, 264,000 stadia would correspond to the terrestrial circumference.

The surface area or capacity of the larger terrestrial circle is investigated thus: the semi-diameter of the Earth is multiplied by half of the circumference, or the ambit of the Earth. Or, the entire diameter is multiplied by the fourth part of the circumference, because the area or capacity of the said terrestrial circle is produced in this way. For Archimedes proves that a rectangle contained under the diameter of a circle and under the extended circumference of the circle is quadruple the area of the circle. Therefore, the diameter of the Earth is nearly 5,727 and 3/11 stadia, which, multiplied by 45,000 stadia—the measure of the fourth part of the entire terrestrial circumference—produces an area of the larger terrestrial circle of 2,577,285,000 stadia. Similarly, if the diameter of the Earth of 115 and 10/11 miles is multiplied by the fourth part of the circumference, namely by 5,625 miles, the aforesaid area of 40,269,375 miles will be found.

Furthermore, the connected surface of the entire terrestrial globe will be obtained by quadrupling the area of the larger circle. Indeed, Archimedes proves in proposition 31 of the first book On the Sphere and Cylinder that the surface of any sphere is quadruple the largest circle within it. The said connected surface will also be produced in another way, namely by multiplying the diameter by the entire periphery of the terrestrial globe; for since Archimedes demonstrated that the rectangle under the semi-diameter of a circle and the fourth part of its ambit is equal to the area of the circle, the rectangle under the diameter and the entire ambit of the circle will therefore be quadruple this, and consequently equal to the connected surface of the globe. Thus, the surface of the entire Earth and water according to these methods is 10,309,140,000 stadia or 161,077,500 miles.

Lastly, the solidity of the entire terrestrial globe of earth and water is defined by multiplying its semi-diameter by the third part of the surface of the entire sphere. For the solid rectangle constructed from the semi-diameter of the sphere and the third part of its ambit is equal to the sphere itself. Or it will be obtained in other ways, which Peucerus recounts in his booklet on the measurement of the Earth. From these, therefore, it is concluded that the solidity of the entire terrestrial globe is 98,405,895,870,000 stadia or 192,192,303,750 miles.

On the fourth chapter.

An ornate drop cap 'I' features scrolling acanthus leaves and floral patterns.

In this chapter, Ptolemy instructs the geographer who intends to describe the entire known world, or even any particular region. He proposes that he should first accept as a foundation those locations that have been observed by excellent celestial instruments, so that the longitudes of these places may be explored through solar or lunar eclipses, and their latitudes through astronomical organs. To these, he must then connect other locations that have been observed either through history, through mere travel, or by any other means; in such a way, however, that the sites and positions of these latter places correspond as closely as possible to their true sites and positions. Ptolemy further testifies that few among the ancients observed the longitudes of places

through eclipses, and he commemorates only one observation of two locations made by a lunar eclipse, which was indeed seen in the town of Arbela in Assyria at the fifth hour of the night, but at Carthage at the second hour. From this observation, the difference in longitude between these two places is clearly elicited, that is, by how many degrees of the equator one recedes from the other toward the East or the West, by assigning fifteen degrees of the equator to each hour. Since the difference in time that intercedes between these two locations is three hours, they differ in longitude by 45 degrees. Therefore, Ptolemy sets the longitude of Arbela at 80 degrees and that of Carthage at 34 degrees and 50 minutes. Pliny cited a similar example in chapter 70 of the second book on the heavens and the world to confirm the roundness of the Earth, when he says that an eclipse of the moon was seen in the town of Arbela in Assyria at the second hour of the night, which was seen in Sicily at the beginning of the night. Since Arbela, as we said above, has a longitude of 82 degrees, the longitude of Syracuse would indeed be 50 degrees if this observation is true, such that the difference in longitude between these two places is 32 degrees, which corresponds to two hours. But Ptolemy notes the longitude of Syracuse at 39 and a half degrees.

However, let us now teach how the longitudes of cities and places are explored. The ancient geographers left us only one way through lunar eclipses, which almost everyone follows as the more certain one. For when the moon is obscured at the same moment of time everywhere—according to different calculations of that same time occurring due to the diversity of meridians—the diversity of the meridians themselves is known. For a lunar defect will be seen in two different places either at the same moment, and then both places exist under the same meridian and there will be no difference in longitude between them; or it will be seen sooner in one place and later in another. And then that place where it will appear sooner, that is, where fewer hours will be counted, will be more easterly than the other. But some things must be heeded, namely that in observing an eclipse, either its beginning must be observed in both places, or its middle, by consulting ephemerides astronomical tables or astrological tables. But it is safer to observe the beginning of the total obscuration or the beginning of the recovery of light in a total eclipse that occurs with duration. Then, the moment of time must be known either by some most accurately crafted clock or, better, by the altitude of some fixed star.

Besides this, other ways are handed down by artists for finding the longitudes of places: that is, either by a clockwork mechanism, or by a portable clock, or by the true motion of the moon, and also by some fixed star that does not have more than five degrees of latitude from the ecliptic, as Werner hands down, and after him Orontius in a peculiar booklet which he inscribed on finding the longitude of places; or also with the aid of a magnet according to its variable declination from the meridian, which method Livius Sanutus treated in his Description of Africa, and Giovanni Battista Porta in his Natural Magic, recently printed. But all these ways are more difficult and obscure, in which a small error produces a very great difference in longitude.

Lastly, the difference in longitude can also be known very easily from a given longitude of one place and the angle of position under which the other place is constituted, along with the itinerary interval of both places, about which we will speak elsewhere. It must not be ignored, however, that there are two beginnings of longitude among geographers: one from the Canary Islands, as in