TITULO VIGESIMO PRIMERO

The useful ways in which Galen may have obtained his information are few. Obtaining his results by use of an accurate water clock can be discounted, as Neugebauer, when writing about the ancient astronomical definition of a year and the determination of the length of a sidereal year argues, that; µSighting instruments could be constructed with fair accuracy whereas sun dials and in particular water clocks were utterly inadequate for the determination of small changes in time intervals.¶14 Another way that Galen may have produced these figures was through the direct measurement of sundials. However since accurate examples of these were produced from geometrical constructions many errors would have been eliminated by examining these figures directly, as will be shown later.

Another possible source for Galen¶s data could have been the works of Ptolemy of Alexandria. The Almagest has been firmly dated to the reign of Marcus Aurelius.15 It is therefore possible that Galen could have seen or heard about them during his stay in Alexandria or at some later time. The relevant data from Ptolemy¶s Almagest and Geographia is set out below.

The first set of data is taken from the Almagest for two of the traditional clima or bands where the data for the length of daylight and for the length of the noon shadow at the equinoxes and solstices was considered to be usefully the same as the data for Alexandria and Rome, that is Lower Egypt and Hellespont.16

14 _{Neugebauer (1975) 54.} 15 _{Toomer (1984) 1.} 16 _{Neugebauer (1975) 41.}

CHAPTER 5:GALEN ON TIME

74 M = the length of the longest day in equinoctial hours

f = the latitude

S1, S2, S3 = the length of the shadow cast by a vertical gnomon of 60 units height at the summer solstice, the equinox and the winter solstice respectively.17

Locality M _f S1 S2 S3

Lower Egypt 14 30;22ƒ 18 6 5/6 35 1/12 83;1219 (30;22) (6;50) (35; 5) (83; 5)

Hellespont 15 40;56ƒ 18 ½ 52 1/6 127 5/620

(40;56) (18;30) (52;10) (127;50)

In Ptolemy¶s Geography the following values are given for the positions of: Rome _{lz go/ ma go/}21 = 36 2/3 41 2/3 = 36;40 41;40 Alexandria _xL_{/ la}22 = 60 ½ 31 = 60;30 31;00

These latitudes are very close to the modern figures.

17 _{In the following table there is a mixture of hexadecimal fractions and ordinary fractions.}

Hexadecimal fractions are indicated by the symbol ; . Thus 83;12 = 83 12/60 = 83 1/5.

18 _{The un-bracketed figures are taken from Toomer¶s translation of the Almagest. Regarding the}

hexadecimal value given at S3 for Lower Egypt, Toomer notes that he is: µReading _{pg ib (}with L) for _{pg ib/} at H108,20. Computed: 83;10,39. Ptolemy does not often use the aliquot fraction _e/ (1/5).¶ Toomer p. 85. n. 39. The figures in brackets are those from the same table translated by Neubauer. The differences do not happen to change any of my calculations but it does illustrate the difficulties in scholastic agreement about these texts. Neubauer comments on this table that: µThe numbers are, of course, the equivalent of a sexagesimal table of tan (_f - _e), tan _f, and tan (_f - _e), respectively, rounded here to 0;5 units because the results are expressed in terms of unit fractions ½ 1/3 ¼ 1/6 and 1/12. The results agree very well with modern computations, usually remaining within 2 or 3 minutes of the given value of _f.¶ Neubauer (1975) p. 45.

19_{Almagest H 108 = Toomer (1984) 85.} 20_{Almagest H 109 = Toomer (1984) 86.}

21_{Geographia III, 1, 61. = Nobbe (1966) I ,151,26.} 22_{Geographia IV, 5, 9. = Nobbe (1966) I , 251,15.}

The following information is given by Ptolemy on the relative daylight in the two cities:

3. Tîn m|n oân xn tÍ 'Italdv xpis»mwn pÒlewn tÕ m|n basdleion `Rèmh t¾n megdsthn ¹myran {cei ærîn cshmerinîn ie ib/ , kag diysthken 'Alexandredaj prÕj dÚseij érv a L/h/: 23

(Rome « has longest days equalling 15 + 1/12 (15;05) this varies from Alexandria where you have a difference of 1 +½ + 1/7 (22/14 § 1;34;17) equinoctial hours.)

These figures produce a total of 13;30;43 (15;05;00 ± 1;34;17) equinoctial hours for the length of the longest day in Alexandria. Neugebauer however takes the longest day at Rome as being 15;12 equinoctial hours.24 _{Since he refers to the same}

information from Nobbe above, he is, presumably, now following Toomer¶s reasoning concerning the reading of _ib/ or 1/12 as _ib or 0;12. This figure then produces a result of 13;37;43 (ie. 15;12;00 ± 1;34;17) equinoctial hours. However neither of these comes at all close to Galen¶s simple figure of 14 equinoctial hours by 29;17 minutes and 12;17 minutes respectively. It is possible that Galen had taken the figure of 14 equinoctial hours for the longest day in the clima of Lower Egypt as his figure for Alexandria. But we then have no explanation for his failure to go on to use the figure of 15 equinoctial hours for the clima for the Hellespont, which, as Neugebauer has said, is approximately correct for Rome.

If Galen had known and used the Vitruvian shadow ratio for Rome at the equinox of 8/9, 25 _{he could have calculated the equivalent Ptolemaic shadow length as 53 1/3}

23_{Geographia VIII, 8,3 = Nobbe (1966) II, 205, 5-9.} 24 _{Neugebauer (1975) 41.}

CHAPTER 5:GALEN ON TIME

76 and made the interpolation between Ptolemy¶s figures for Hellespont and Massalia and come up with his µlittle¶. But if he makes the correction here why does he not make a similar correction for Alexandria? In this latter case the Vitruvian shadow ratio for Alexandria is 3/526 _{which produces an equivalent Ptolemaic shadow ratio}

of 36 against his ratio for Lower Egypt of 35 1/12.

Toomer states that: µExcept where it is necessary to be precise, Ptolemy prefers the traditional Greek fractional system to the sexagesimal¶.27 Goldstein states as well that:-

In Greek geography the latitude of cities is not given in terms of local altitude of the celestial north pole above the horizon, but as a ratio of longest daylight to shortest night, longest daylight measured in equinoctial hours, or the ratio of gnomon to shadow on a sundial for solstice or equinox. Moreover, where ratios are given, there is a strong preference for ratios of small integers, e.g., 5 to 3 or 7 to 5.28

It has already been noted that Neugebauer also thinks that Ptolemy prefers simple fractions. These become very important in the examination of the geometrical solutions that follow. Before that, however, it is necessary to understand the manner in which such a solution may be found using geometrical techniques. The best surviving description of the geometrical construction used is by Vitruvius. However he only leaves a description of the analemma itself without any instruction as to its use. While I would think that other people have examined this problem, I have not been able to find a monograph that has actually described in detail how it

26_loc.sit.

27 _{Toomer (1984) 7.} 28 _{Goldstein (1985) 4.}

could be used. In the following section I will do so as it is important to an understanding of Galen¶s comments about the geometrical determination of the shape and markings of sundials.