Fig. 1 is the Planisphere adjusted for the northern latitude of 30° 22′ (where the longest day consists of fourteen equatorial hours), agreeably to the “Exemplification” given by Ptolemy in Chapter XV, Book 3. It represents that portion of the celestial sphere which is contained between the tropics: the central horizontal line is the equator; the curved line extending longitudinally from east to west is the ecliptic; the central perpendicular line is the meridian, or cusp of the 10th house; the other short lines, cutting the equator transversely, are the cusps of the other houses; that of the 1st house being the eastern horizon; that of the 7th, the western horizon. Hence, the distance from the 1st house to the meridian, or from the meridian to the 7th house, shows the semi-diurnal arc of any parallel of declination in the ecliptic; and the distance of the 7th house to the 4th, or from the 4th to the 1st, shows the semi-nocturnal arc. The distance from the cusp of one house to that of the next, taken on the same parallel, is also equal to two temporal hours; thus, for instance, in the latitude above quoted, the semi-diurnal arc of 0° ♊ is 6 h. 50 m., or 102° 39′ of the equator; consequently the diurnal temporal hour is equal to one equatorial hour and eight minutes, or to 17° 6′ of the equator. In his first example, Ptolemy directs 0° ♉ to be placed on the ascendant, so that the beginning of ♑ may be on the mid-heaven; 0° ♊ must, therefore, fall on the point A, distant from the mid-heaven 147° 44′ of the equator, as measured by the line AB; because every point in the sphere always preserves one and the same parallel with the equator; and 0° ♊, in passing to the mid-heaven, must proceed along the line AB. In the present case, however, it is required to know how long 0° ♊ will be in coming to the ascendant, the given position of 0° ♉. Now 0° ♊ will be on the ascendant when it arrives at the point G; therefore the distance from A to C is the amount of the prorogation between 0° ♉ (when posited on the ascendant) and 0° ♊, and it is equal to 45° 5′ of the equator. In the second example, 0° ♉ is placed on the mid-heaven, which position must be at D, so that 0° ♊ must necessarily be at E; and the distance from E to B, equal to 57° 44′ of the equator, is the prorogation between 0° ♉ and 0° ♊, when 0⁴ ♉ is on the mid-heaven. In the third example, 0° ♉ is supposed to be on the 7th house, descending, at F, so that ♓ is on the mid-heaven, and 0° ♊ at the point G, in advance of the mid-heaven 32° 16′ of the equator, as shown by the distance BG. Now it is required to bring 0° ♊ to the 7th house (the place of 0° ♉), and it will be there on arriving at H, distant from B 102° 39′ of the equator; but as 0° ♊ is already at G, the distance from G to H, equal to 70° 23′ of the equator, is the amount of the prorogation between 0° ♉ and 0° ♊, when 0° ♉ is on the 7th house. The fourth example places 0° ♉ at I, three temporal hours past the