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

Moreover, I believe the promontory of Cory is not the modern Cape Comorin, as some wish it to be, but rather Cape de Colle, as is pleasing to Mercator. Indeed, Cape Comorin seems to me to be the extreme Commaria of Ptolemy. However, Curura seems to me to be the modern Cumura, due to the similarity of the name and the location.

The second navigation, which Ptolemy criticizes and corrects, is reported from Curura to Paluras towards the winter sunrise, that is, towards the southeast, at 9450 stadia. Let the city of Curura be A in the underlying figure, and Palura be B, declining 30 degrees from the parallel AC drawn through Curura: for the southeast wind declines 30 degrees from the equator towards the south. Let the navigation along the meridian through Palura be CB, and let the navigation from Curura to Palura be made along the direct path AB, and not along a winding path. Since, therefore, this navigation is supposed to be along the line AB

A diagram depicts a right-angled triangle labeled with vertices A, B, and C. Vertex A is on the left. Vertex C is on the right, aligned horizontally with A. Vertex B is positioned directly below C. The horizontal line is AC, the vertical line is CB, and the hypotenuse connects A and B.

of 9450 stadia, if a third part of this number, namely 3150, is subtracted because of the inequality of the navigation, there remains the direct path AB of 6300 stadia. Since the angle CAB is one-third of a right angle, the remaining acute angle ABC will be two-thirds of a right angle, for the angle C is a right angle, as was evident above in the other demonstration. Hence, as above, the line AB will be double the line BC, and consequently the square of side AB will be quadruple the square of side BC. But the two squares of AC and CB are equal to the square of side AB; therefore, said two squares AC and CB will be quadruple the square of CB. Thus, of such parts as the square of AB is four (which is the two squares of AC and CB), the square of CB is one; and therefore the remaining square of the line AC will be three, of such parts as the square of AB is four. Wherefore the ratio of the square of side AB to the square of side AC will be four-thirds, that is, as the square number 36 is to the square number 27. But the square root of the number 36 is 6, and the square root of the square number 27 is nearly 5. Therefore, the ratio of the direct path AB to the portion of the parallel AC is as 6 is to 5. Wherefore, from the 6300 stadia of the direct path AB, a sixth part

being subtracted, there remain 5250 stadia of the parallel AC, which are to be converted into ten and a half degrees by dividing them by the number 500. For since such a parallel is distant only 12 degrees from the equator, it differs little from the magnitude of the equator, and 500 stadia can be safely taken for each degree, just as in the equator.

But the said arc AC of the parallel drawn through Curura can also be obtained by another method, assuming a rectilinear triangle ACB, since those portions of the arcs do not differ much from straight lines. For since the side AB of 6300 stadia is given, its square will be 39,690,000. Now, side CB is half of side AB, as was concluded in the previous example, namely 3150 stadia, the square of which is 9,922,500. Subtracted from the square of said side AB, there remains the square of side AC, 29,767,500, the square root of which, taken from our Tetragonal table, will give the side AC of nearly 5456 stadia. Which method is more exact than the Ptolemaic one. We can also explore the same side AC by the thirteenth proposition of our Plane Triangles by means of sines. The cause of the diversity, however, is that Ptolemy supposes the ratio of side AB to side AC to be as 6 is to 5, and the true ratio is as 62 to 52. Whence, if we multiply the number of stadia of side AB by the number 52, and divide the product by the number 62, we will collect 5460 stadia. This calculation differs little from our other calculation.

Moreover, Palura is today Cincapura modern Singapore, at the tip of the Malaccan Chersonese, which almost touches the equator. This does not recede 30 degrees from the parallel of Curura, as Ptolemy noted, but nearly an eighth part of the circle.

The third navigation, which Ptolemy reviews from Marinus, is from Palura to Sada towards the equinoctial sunrise, of 13,000 stadia. Between these cities intervenes the Gangetic gulf, which is posited at 19,000 stadia. Because he believed this navigation to be false, being a nearly direct journey under the same parallel, Ptolemy subtracts only a third part on account of the irregularity of the navigation, so that the corrected distance between the two places is left as 8670 stadia, to which 17 and one-third degrees are owed. Here the error is clear, by which Ptolemy, with Marinus, designed such a different face of this eastern part of the earth that it has almost no similarity with the description of the moderns. Therefore, the cities of Palura and Sada, between which the Gangetic gulf intervenes, are not situated in the same parallel distant 11 and a half degrees from the equator as Ptolemy posited, nor was the navigation made towards the equinoctial sunrise, but rather towards the north, which wind recedes from the equinoctial sunrise by 60 degrees towards the north. Hence, Sada will be in a parallel distant from the equator where Palura falls, by 28 degrees. For we believe the city of Sada to be the city called Zaiton today, or at least that it was near it. Nor should it seem strange that Ptolemy was deceived with Marinus in the description of this part, since he had constructed it from the reports of merchants who sailed to these places. That the Gangetic gulf is now between Cincapura and Zaiton is clearly argued by the islands placed by Ptolemy in the said gulf, which agree with the description of the moderns. For the island of Bazacata in Ptolemy is for us today Palohan, the island of Good Fortune in Ptolemy is for us Borneo, and the Barussae islands of Ptolemy are today called by us the Philippines, the more recent names of which are Cebu, Cailon, Mindanao, Tandair, and Luzon.