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

Since, therefore, the axis DB passes through both the center D of the great circle AEC (namely the Equator) and through the center I of the parallel FGH, and stands at right angles to its plane, and makes both of the adjacent angles FIB and FID right angles, therefore by the definition of the right sine, the straight line FI emanating from the center I of the parallel FGH will be the first right sine of the arc FB, which is the complement of the known arc AF (the distance of the parallel from the Equator). Since the ratio of circles between themselves is the same as that of their diameters or semidiameters, as we said in the second chapter, the proportion of the entire circumference of the Equinoctial AEC to the entire circumference of the parallel FGH will be that of the semidiameter AD of the Equator to the semidiameter FI of the parallel. But the straight line AD is the whole sine of the entire quadrant ABD, and it is always given as 100,000 parts; therefore, FI will also be the first right sine of the known arc FB, since it is the complement of arc AF. But as an example, let it be proposed to find the proportion of the parallel FGH drawn through Thule to the Equinoctial AEC. Since this parallel is distant 63 degrees from the Equator, arc AF will be 63 degrees, therefore arc FB, its complement, will be given as 27 degrees, whose sine in the tables is 45,399 parts. Therefore, the ratio that the whole sine AD (100,000) has to the sine FI (45,399) is the same that the entire circumference of the Equinoctial AEC has to the entire circumference of the parallel FGH. In order to know if there is a proportion between these circles like the number 115 to the number 52, as Ptolemy supposes, we will proceed in this manner: we will multiply 115 by 45,399 and divide the product by 100,000, for the number corresponding to said parallel will emerge in the quotient. From the multiplication of the number 45,399 by the number 115, 5,220,885 arises, which, divided by the whole sine, gives 52 and a little more in the quotient. Wherefore, the proportion of the parallel through Thule to the Equinoctial as 52 to 115 is correctly assigned by Ptolemy. By what reasoning Ptolemy assumed these numbers, we will see later below in the last chapter. Similarly, let us now examine if the proportion of the Rhodian parallel to the Equinoctial is truly as 93 to 115. Since such a parallel is distant 36 degrees from the Equator, the sine of the complement of this arc is 80,902 parts, by which, when the number 115 is multiplied, the number 9,303,730 emerges, which, distributed into the whole sine, gives 93 for the number that agrees with the parallel. Whence the proportion of such a parallel to the Equinoctial as 93 to 115 is correct. Wherefore it is clear from this that Werner was hallucinating in the sixth annotation of this chapter, and unjustly criticized Ptolemy as being not much practiced in more exact numeration, as he himself says.

Furthermore, this rule is most easily confirmed by a table containing the proportion of any parallel to the Equinoctial, such that the Equinoctial is assumed to be 360 parts, and also another table which exhibits the proportion of one degree of a parallel to one degree of the Equator. For having found the sine of any parallel, we will multiply it by 360 degrees and divide the product by the whole sine, and whole degrees will be gathered; again, we will multiply the remainder from the division by 60 and divide by the whole sine, so that the first minutes emerge, and again we will multiply the remaining number by 60 and divide by the whole sine, so that we have seconds. As in the example above for the parallel of Thule at 63 degrees, whose sine was seen above as 45,399: if this sine is multiplied by 360, the number 16,343,640 will arise, which, divided by the whole sine, will give 163 degrees. The remainder, multiplied by 60, is 2,618,400, from which, divided by the whole sine, 26 first scruples minutes are collected, and the remaining number multiplied again by 60 and divided by the whole sine produces 11 seconds. Therefore, the total arc of that parallel is 163 degrees, 26 minutes, 11 seconds, of such parts as the Equinoctial is 360.

Furthermore, if we multiply the aforesaid number 45,399 by 90 and divide the product by the whole sine, and maintain the same order as above, we will gather the number of degrees of the quadrant of this parallel, of such parts as the quadrant of the Equinoctial is 90. For in this operation, the number 4,085,910 is first formed by multiplication, which gives 40 degrees. From the remainder, however, multiplied by 60, 51 minutes and nearly 33 seconds are gathered. Wherefore the quadrant of this parallel of Thule has a ratio to the quadrant of the Equinoctial as 40 degrees, 51 minutes, 33 seconds to 90 degrees. Although in Werner's table 40 degrees, 53 minutes, 3 seconds are written.

In no dissimilar way will you be able to construct a table embracing the ratio of individual degrees of parallels to individual degrees of the Equinoctial, by multiplying, namely, the number 45,399 by 60 and dividing the product by the whole sine; for you will gather first minutes, then seconds and thirds. Whence for a single degree of the said parallel, there correspond by this reasoning 27 minutes, 14 seconds, 22 thirds, of such parts as one degree is 60 minutes. Werner's table, however, has 27 minutes, 15 seconds, 24 thirds.

On the Twenty-first Chapter.

In this chapter, the author reviews a certain method of description of the world on a plane, in which, indeed, all meridians are designed as straight lines and the parallels as circumferences of circles, yet in such a way that the principal parallels at least—namely those that are drawn through Thule and through Rhodes, and also the Equinoctial—have the due proportion and commensuration between themselves. But since Ptolemy himself teaches this method plainly in the last chapter, we will therefore dwell more upon it there.

On the Twenty-second Chapter.

Ptolemy here hands down a method of describing the world of lands on a spherical surface aptly and neatly, which is sufficiently clear. But since a smaller part of the world was known in Ptolemy's time, and now almost the entire world has been explored, it is therefore necessary to provide the descriptions of the moderns. First, therefore, having prepared a globe of convenient size, we will find two poles on it, passing an iron axis through them which extends beyond the sphere itself near each pole, to the extremities of which axis we will attach a semicircle or an entire movable circle crafted from copper or another metal, which is divided from pole to pole into 180 equal parts, and where the semicircle is divided into equal parts, understand there the position of the Equinoctial, from which you will count 90 degrees on either side toward the poles, marking the numbers at every five or ten degrees, for example. Afterward, by means of this circle, you will first describe on the globe the Equinoctial circle by equidistance from each pole, which you will distribute into 360 equal parts, and at every five or ten degrees you will note the numbers, proceeding from the West toward the East, and where you have placed the beginning of such a computation, understand there to pass the first Meridian, which is drawn through the Canary Islands.