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

of the same Ptolemy. Similarly, Pliny also offers effective arguments for proving the roundness of land and water in Book 2, from chapter 64 to chapter 72 of his work on the heavens and the world. The same author also proves in chapter 69 of that same book that the Earth occupies the center of the universe. Furthermore, in the short work on the sphere by Io. de Sacrobosco John of Holywood, and among its commentators, these matters are treated with sufficient diffusion, and we refer the reader to those authors. We note, however, that when we say the globe of land and water is round, we should not understand it to be perfectly round like the heavens or some globe turned on a lathe. For the heights of mountains, the depths of valleys, and the plains of fields, which are on the surface of the Earth, render the surface of the terrestrial globe truly unequal with respect to us. Nevertheless, when compared to the immensity of the Earth, all these things do not prevent such a globe of land and water from being round, and appearing as such in accordance with celestial appearances. For indeed, all straight lines that would be drawn from the center of the Earth to any part of its surface would be nearly equal, and would differ from one another imperceptibly. Moreover, an exact spherical surface of the Earth was not even necessary, as it is for the heavens, because the Earth persists, and the heavens move continuously and assiduously around it. That the surface of the Earth, if it were to be intersected, as Ptolemy says, by some plane drawn through the center of the terrestrial globe, the line that is on the surface of the Earth at the common intersection would be the circumference of a greater circle, is demonstrated by Theodosius Tripolita Theodosius of Bithynia in the first proposition of the first book of his Spherics.

Furthermore, if Ptolemy also carries these lines up to the concave surface of the heavens, these lines will indeed encompass two arcs similar to one another, of which one will be of the celestial circle and the other of the terrestrial, and each of these will have the same proportion to its entire circle as the other has. We will confirm this with the following demonstration.

A geometric diagram depicts two concentric circles. The smaller inner circle represents the Earth, and the larger outer circle represents the celestial sphere. Both share a common center, B. Radial lines B A and B C extend from the center through points D and E on the inner circle to points A and C on the outer circle. The diagram illustrates how radial projection creates similar arcs on both the celestial and terrestrial spheres.

Let there be two circles in the same plane, namely E D G on the surface of the Earth, and C A F on the concave surface of the heavens. Let two lines, B A and B C, be drawn from the center of the Earth B out to the celestial circle. I say that these lines encompass two similar arcs, namely E D and C A. That is, the proportion of arc E D to its entire terrestrial circle E D G is the same as that of arc A C with respect to the celestial circle C A F. Since, therefore, the angle E B D to four right angles, which encompass the entire circle, has the same proportion as the arc E D, opposite said angle, has to the entire circumference E D G according to the second corollary of the 33rd proposition of the sixth book of Euclid's Elements, and since the angle C B A, which is the same as the angle E B D, also has the same proportion to those same four right angles as the arc C A has to the entire circle C A F according to the same corollary, therefore, by the 11th proposition of the fifth book of the Elements, the arc E D will have the same proportion to the entire arc E D G,

as the arc C A has to the entire arc C A F. Wherefore the said arcs E D and C A are similar. This is what we proposed to demonstrate. This same conclusion can be reached regarding two equal angles that encompass similar arcs of two unequal circles.

Regarding the third chapter.

"As there were before us not only a straight distance, &c." In this third chapter, Ptolemy first hands down the method observed by the ancient Geographers for finding the circumference of the entire terrestrial globe, by means of the direct distance between two locations lying under the same Meridian, which is as follows. Let one observe in one place the elevation of the pole, either by a Quadrant, as we taught in the seventh and eighth propositions of the fifth book of our own work on the use of the Quadrant, or even by any other instrument. Then let one travel under the same Meridian in a straight journey, until one knows the elevation of the pole to be greater or lesser by one degree, or a half, or another sensible quantity. For in progressing under the same Meridian, the elevation of the pole necessarily varies continuously. Let one subtract the lesser elevation of the pole from the greater, so that the difference of the elevation of the pole between those two places, or if you prefer, the difference of the vertical points of both places, becomes known. Let one also measure the interval between both places, so that the stadia units of distance or miles of the journey between them may be known. Then one will multiply this distance by the number of degrees of the entire circle, namely by 360, and divide the product by the polar difference. For in this way, the number of the circumference of the entire terrestrial globe will arise. But for a clearer understanding of this matter, let the circle A C F in the following figure be the celestial Meridian, and let the circle D E G be the greatest terrestrial circle lying beneath the celestial Meridian itself in the same plane. Let A be the vertical point of the first place D on Earth, and let C be the vertical point of some location existing at E. Let the journey between the said two locations D and E then be observed, to which corresponds the arc A C on the Meridian, and the variation of the points A and C. I say, therefore, that the aforementioned terrestrial distance D E has the same proportion to the entire circumference of the terrestrial circle D E G as the celestial arc A C has to the entire Meridian A C F, because the arcs D E and A C are similar, as we demonstrated at the end of the preceding chapter. There are three known numbers here, namely the arc A C, that is the polar difference or the difference of the vertical points, the arc D E, that is the itinerary distance of the two locations D and E, and the entire arc A C F of 360 degrees. Hence, from the rule of three, the entire arc D E G will also be determined. By this method, Ptolemy discovered the circuit of the entire Earth to be 180,000 stadia. For he sets down this many stadia in the seventh chapter of the fifth book of this volume, and these indeed make 22,500 of our miles, or 5,625 German miles. For he finds that to each degree there correspond 500 stadia, which make 62 and a half Italian miles, or 15 German miles with five eighths of one mile.

A circular diagram depicts two concentric circles centered at B. The outer circle represents the celestial meridian (A C F), and the inner circle represents the Earth (D E G). Points D and E on the Earth correspond to points A and C on the celestial sphere. Radial lines extend from B through D to A, and through E to C, illustrating the relationship between terrestrial distance and celestial latitude.