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

500 stadia, which make 62 and a half Italian miles, or fifteen German miles with five eighths of one mile.

Eratosthenes, as Cleomedes recites in the first chapter of his first book on the magnitude of the Earth, investigated by another way, little different from this one, that the terrestrial circumference contains 250,000 stadia. For he observed the variety of the meridian shadow that occurs in two places situated under the same Meridian, which are Alexandria in Egypt and Syene, which two places he found to differ by 50 parts of the celestial circle and to be distant by 5,000 stadia, which stadia multiplied by 50 give a terrestrial circumference of 250,000 stadia. Pliny, however, in chapter 108 of the second book on the heavens, puts a greater sum of stadia according to the opinion of Eratosthenes himself, namely 252,000, because he assigns 700 stadia to each degree of the meridian, although according to Eratosthenes there are only 694 with four ninths of one stadion. The same Pliny says that Hipparchus adds nearly 25,000 stadia to this number of Eratosthenes regarding the circumference of the Earth. Whence, according to them, the terrestrial circumference will be 277,000 stadia. There are, however, those who wish there to be no true difference between these authors, except that the stadia of Hipparchus are somewhat smaller than the stadia of Eratosthenes. Hipparchus attributes 774 stadia to each celestial degree.

Furthermore, the circumference of the entire Earth can be found from the observation of the meridian altitude of some known fixed star. For if two locations were taken under the same Meridian, the itinerary distance of which is known, and then the meridian altitude of the same star was observed in both locations, the entire circumference of the entire Earth will become known from the difference of the meridian altitude of the star that arises at each location, and from the given itinerary distance of said locations. By this way, Posidonius investigated the circumference of the Earth to be 240,000 stadia, as Cleomedes recites in the same place cited above. For he chose two cities subject to nearly the same meridian, namely Alexandria in Egypt and Rhodes, in which he observed the diversity of the meridian altitude of the most splendid star in the ship Argo, which is commonly called Canopus. He observed that this star, which lay hidden permanently concealed under the horizon of Greece, began to show itself to be seen by those progressing toward the south at the horizon of Rhodes, yet only for a small moment, as if brushing the horizon, and soon submerging beneath it. But in the city of Alexandria, as one recedes further toward the south under the same meridian, the same star shines for a longer duration and is raised higher above the horizon, specifically by the 48th part of the entire meridian, or 7 and a half degrees. Whence, as much as the arc of the meridian from Canopus to the horizon of Alexandria is with respect to the celestial meridian, or the entire celestial circle, so much is the interval of Rhodes and Alexandria, or the portion of the terrestrial greatest circle drawn through both locations with respect to the greatest terrestrial circle itself. But that interval was measured to comprise 5,000 stadia, which, if multiplied by the number 48, will produce a circumference of the greatest terrestrial circle of 240,000 stadia.

I will not omit to say that Francesco Maurolico also hands down another method in his dialogues of Cosmography for knowing the terrestrial circumference, which is indeed very ingenious and subtle; nevertheless, it has its own difficulties due to the weakness of our sight, which can with difficulty discern from any high mountain the final limit of the seen Earth, which is indeed required in that task.

"Whence if one does not measure the circle through the poles, &c." Ptolemy asserted in these words that even if two locations, whose itinerary distance is known, have not fallen under the same Meridian, the circumference of the terrestrial circle can nonetheless be explored. But according to this method, it is necessary to know the portion of the greatest circle intercepted between those two locations. This will indeed be done if we have explored the elevation of the pole of both locations and the angle of position by which one location is distant from the meridian of another. For thus a spherical triangle will be conformed, of which two sides will be known as the complements of the latitudes of the two assumed locations to a quadrant, along with the angle comprised by them, which is called the angle of position, and it determines the difference of longitude of both locations. For from these given things, the third side in the said triangle will be known, by which the two locations differ from one another according to a portion of a great circle. But we will teach this method in our short work on Spherical Triangles, which we will soon bring to light, in which we will treat fully and with a new method regarding such triangles, using certain tables fashioned by us, in which, by a solitary lateral entry, both the sides and angles of all spherical triangles will be able to be obtained most easily and most promptly. In the same place also, whatever pertains to the Geographical task will be abundantly handed down. Furthermore, we will also publish the Quadratum Meteoroscopium Meteoroscopic Square in the same volume, whose use will be most delightful not only for promptly eliciting the dimensions and calculations of all spherical triangles, but for completing all Geographical operations and all those of the entire primum mobile first moved sphere.

"The same will be patent to us if through an instrument which more sublime, &c." Ptolemy affirms that the aforementioned, and many other things besides, can be done most easily with the help of a certain Meteoroscopic instrument elaborated by him, the construction of which handed down by the same author we do not possess. Regiomontanus, however, showed that he had investigated it, and he handed down the composition of a certain Meteoroscope built in the form of an Armillary sphere, and he teaches through it certain not unpleasant things regarding the latitudes and longitudes of locations and regarding terrestrial dimensions, as can be seen in his own work.

It does not seem alien to my purpose to teach now how, from a known terrestrial circumference, we can investigate the diameter or depth of the same Earth, the area or capacity of the entire greater terrestrial circle, how great is the convex surface of the entire terrestrial globe, and finally how great is the solidity of the same globe or ball.

The diameter, therefore, or the depth of the Earth is investigated thus. Subtract the twenty-second part from the number of the circumference of the Earth, and divide the remaining number by 3, and what will come forth will be the diameter of the globe itself. For Archimedes demonstrates in his booklet on the measurement of the circle that the proportion of the circumference to its diameter is nearly three and one-seventh, that is, as 22 is to 7. Whence the terrestrial circumference could also be multiplied by 7 and the product divided by 22, for the same number of the diameter will also be collected. Therefore, since from Ptolemy the circuit of the Earth is 180,000 stadia, multiply this number by seven and the number 1,260,000 arises, which, divided by twenty-two, gives the diameter of the Earth as 57,272 and eight elevenths stadia, which make 7,159 and a half miles; the radius will be 28,636 and four elevenths stadia, which make 3,579 and nearly a half miles. Conversely, however, if one has the known diameter of the Earth, as happens in the method handed down by Maurolico, one will multiply it by 22 and divide the product by 7, for thus the entire terrestrial circumference will arise.
Pliny writes in chapter 109.