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Figure 4.3. A 15th-century copy of a Chinese horizontal sundial from a design by Guo Fhoujing, Yuan Dynasty, 1279-1368; now located in the courtyard of the Old Beijing Observatory in Beijing, China. Photo by E.F. Milone.

rises earlier and sets later than it would on an airless world, a correction must be made to increase the computed hour angle. With an assumed refraction correction of 34 arc-minutes, the range of effects on H for the sites given in Table 4.1 is 0.110 hour for Site 1, to 0.041 hour at the equator and +0.046 hours at the tropics of Cancer and Capricorn. An additional correction should be taken for the radius (the "semidiameter") of the solar disk, because daylight may be reckoned from the first gleam of the Sun, rather than the appearance on the horizon of the center of the disk. The total correction to the HA of rise is dH ~ 1.5 x (34/3437.7)/(cosf-cos8©-sinHr), which, by symmetry, increases the setting hour angle by the same amount. Thus, the correction is added to both summer and winter HA values before the ratio is taken. The resulting corrected ratios are 2.875 for Callanish (Site 1), 1.511 for Rhodes, 1.404 for Alexandria, 1.272 for the tropics, and 1 for the equator.

The interval of time between the rising of a sequence of the zodiacal signs permits the number of hours of the night and, thus, the number of hours of daylight to be obtained. The correlation of the ratio of daylight to night hours with latitude permitted the ratio, or, alternatively, the number of degree-hours to be used to specify the latitude band or "climate" in which a site was located. There were traditionally seven "clima" (Greek Kli|a). Babylon had a ratio of longest to shortest day lengths of 3:2, and methods were devised ("System A" and "System B") to determine this ratio. Although the inventor of the interval ratio as a latitude indicator is unknown (Neugebauer 1975/1969, pp. 184-185), these ideas were being applied in Alexandria by Hypsicles in the 2nd century b.c. At this time, Alexandria

Figure 4.4. An example of a modern (1968) vertical sundial in Lucerne, Switzerland: The zodiacal constellations are prominent. Photo by E.F. Milone.

Figure 4.4. An example of a modern (1968) vertical sundial in Lucerne, Switzerland: The zodiacal constellations are prominent. Photo by E.F. Milone.

Figure 4.5. The style of a vertical sundial from Greco-Roman Egypt, found at Luxor. Drawing by Sharon Hanna, after Borchardt (1917).

was in the first clima, "3,30" (= 14h 00m for the summer maximum of sunlit hours)5 and other zones were defined by adding multiples of 4°, the second at 3,34, and so on, to 3,54. In this scheme, Babylon was squeezed out of the "clima," but another scheme existed in parallel in which Babylon appeared in the second clima, with 3,32, with the third clima at 3,36, the fourth at 3,40, and soon to the seventh at 3,52. Several centuries later, Ptolemy (Almagest, Book II), describes 20 parallels of geographic latitude beyond the equator, separated by quarter-hour intervals. He tabulates 10 of these zones (Book II, §13) in half-degree intervals. The 10 zones are designated according to the maximum hours of

5 "3,30" is to be read (3 x 60) + 30° = 210° (or 210/15 = 14 hours).

Figure 4.6. A stone equatorial sundial in the Forbidden City in Beijing, China: The dial disk lies in the plane of the celestial equator and thus faces the north celestial pole. Photo by E.F. Milone.
Figure 4.7. A sundial at the Jaipur observatory of Maharajah Jai Singh, dating from the 18th century. Photo by E.F. Milone.

sunlight and are separated by multiples of a half-hour. The first zone beyond the equator was 12,30, the second 13,00, to a maximum of 17h. Within this scheme, the first "climate" is at Meroe, near modern Asuan, with 13h. There is no mention of a refraction or semidiameter correction in Book II of the Almagest. Toomer (1984, p. 421, fn. 8) says that the only reference to the effect of refraction (if that is what Ptolemy is talking about) is given in Book IX, §2, where, referring to the times of visibility or invisibility of planets, Ptolemy says, "but the times too can be in error, both because of atmospherical differences and because of the differences in the [sharpness of] vision of the observers" (the bracket is Toomer's). In addition to timings and arcs of the celestial equator, however, Ptolemy refers to the use of a gnomon to establish shadow lengths and uses the ratio of the meridian shadow length at winter solstice to that at the equinox (Book II, §5; Toomer 1984, pp. 80-82).

Table 4.1. Seasonal variation of day length at various sites.
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