¹ Seneca attributes the majority of errors in astrological predictions to the fact that they were based on just the five planets and did not take adequate account of the fixed stars, NQ 2.32.7. The fragments and testimonia concerning Khairemon, one of the earliest surviving authors on ancient astrological ideas [Cl A.D., contemporary with Seneca], have been collected and translated into English recently by van der Horst 1984.

² Bowen 1999 argues that predicting eclipses in a precise way – when, where, duration of occultation – was not part of the astronomer’s self-defined tasks until the Cl A.D. He sees it growing out of a literary topos. I would look also to the growth of astrology, for this is approximately the same period in which personal horoscopy developed.

³ Thompson made what he called ‘a somewhat startling’ claim that ‘very many of [ancient statements concerning animals] deserve not a zoological but an astronomical interpretation’, 1895, p. xii.

⁴ Hipparkhos observed that even for professional astronomers large errors could arise from the instruments used to measure it, apud Ptolemy Almagest 3.1: ‘In the case of the solstices, I have to admit that both I and Archimedes may have committed errors of up to a quarter of a day in our observations and calculations [of the time]’, Toomer trans., emphasis added. Ptolemy’s attribution of such errors to the ‘construction and positioning of the instruments’ slightly precedes this quote.

⁵ Principally gnomons and sundials, which could be flat or hemispherical, fixed horizontally or vertically, or portable.

⁶ Similarly most water clocks were devices to apportion time, on the basis of the time it took for the water in the clepsydra to run out of the hole at the bottom, rather than devices to tell the time of day. Their use in the courts, for example, regulated the duration of speeches: the clock was started when the speaker started, plugged during the reading out of laws, and the speaker had to finish when all the water had run out. In like fashion, Herophilos’ portable clepsydra ‘measured’ the patient’s pulse rate.

⁷ See Geminos, Elements of Astronomy 8. One of Aristophanes’ complaints in the Clouds (production date 423) is that the monthly feasts and festivals are falling on days which do not correspond with the appearance of the moon. He is assuming that the gods of Olympos work with the lunar calendar, which we may thus suppose to be the traditional one. In the Athens of his day this has been interfered with by political adjustments or set aside in favour of a luni-solar or solar calendar, long enough for a very obvious asynchronization to have developed, and recently enough for Aristophanes to make an issue of it. Meton had offered to sort out the calendar in Athens nine years before Aristophanes produced the Clouds (see Diod. Sic. 12.36.2, recording public honours voted to Meton). He came up with his eponymous Metonic cycle of 19 years, and the first cycle started on 13 Skirophorion (the summer solstice) in 432 B.C. This is usually said to have been used only by the astronomers, and did not become the basis for a reformed civil calendar, but it does not follow from this that it was not used for occasional corrections to the civil calendar. Diodorus says that ‘even down to our own day, the majority of the Greeks (of - ख़ृ π пХеїатоі τών Έλλήνων) use the nineteen-year cycle’ 12.36.3. Diodorus may not be the best historian from antiquity, but he did his research at the Library at Alexandria and had access to much material which has been lost to us. Corrections to the calendar in our own time may deal with seconds rather than months, but such correction is still necessary and is necessarily sporadic. Meton is actually a character in Aristophanes’ Birds (produced 414), and does geometry at lines l000ff. On his water clock see Bowen and Goldstein (1989) p. 77.

⁸ Such adjustments could involve dropping or adding months, not a mere 11 days (which caused so much upset for Pope Gregory’s contemporaries). In Athens intercalations were made by the archon, and the calendar so adjusted was the one known as the archon’s calendar. He made adjustments by moving the peg in the official parapegma (labelled board) the required number of slots forward or backward; on parapegma see Neugebauer 1975 vol.1 pp. 587-9. There were two other calendars in operation in Athens simultaneously: 1. the prytany calendar, of 10 months of 36 or 35 days each (Ath. Pol. 43.2), by which the ten tribes of Athens took turns to be presidents of the state; 2. the kata theon (‘according to the god’) calendar, the nature of which is disputed. See Pritchett 1963 esp. §4.

⁹ Pritchett 1963 p. 339.

¹⁰ Dedicated in 10/9 B.C. Some important remaining parts were found and published by its excavator, Buchner 1982. Besides being emperor, Augustus also had responsibility for keeping the civil calendar in step with the seasons (qua Pontifex Maximus), which at this date meant fine-tuning Caesar’s system. This monumental sundial, next to the Ara Pacis, combined time-keeping and ideological (religious as well as political) functions. The ideological function was obvious to Cicero when Caesar first introduced this calendar: ‘the heavens have to obey the dictator’, Plutarch, Caesar 59.

¹¹ Astronomy was part of the formal education of the Roman agrimensores, on which see Dilke 1971 chapter 4.

¹² Besides optimizing solar gain in winter and cool shade in summer for the residents of the building, the importance of correct aspect for storage rooms of different products is stressed by most of the agricultural writers. See e.g. Columella on the olive-oil store.

¹³ For example, Neugebauer 1975 or Barton 1994.

¹⁴ E.g. Samuel 1972.

¹⁵ See n. 18 below. Eudoxos had produced a (lost) description of the heavens, which Aratos versified, but its style and content must have been very similar to Aratos’ surviving Phainomena. Hipparkhos considered them together in his Commentary on the Phainomena of Aratus and Eudoxus, and says explicitly ‘Now Eudoxus gives the same collection of phenomena as Aratus, but has set them forth with greater knowledge ... It is, perhaps, not fair to blame Aratos if in some points he is found to be in error; for in writing his Phainomena he has followed the arrangement of Eudoxos, without making any observations on his own account, and without professing to be speaking with the authority of a mathematician when giving details of celestial happenings which afterwards prove to be inaccurate . . . that Aratos followed Eudoxos’ account of the phenomena may be gathered by comparing, at length, Eudoxos’ text with that of the poem dealing with the same topic in each case . . . they practically agree in all but a very few details’ [four examples follow], Heath trans. 1932 pp. 117-18.

¹⁶ Eudoxos’ work survives only in fragments, bits quoted or paraphrased by later authors. For discussion of his ideas and contribution to Greek astronomy, see Neugebauer, 1975, vol. 2 pp. 675- 83. For Euclid’s Phainomena see Health 1921 vol. 1 pp. 348-53 and 440-1. Both are excerpted and translated in Heath 1932 pp. 65-7 and 96-100 respectively.

¹⁷ On which see Neugebauer 1975 1 pp. 277-88.

¹⁸ Comm. on the Phainomena of Aratos and Eudoxos 1.4.1 : ‘In fact there is no star at the pole but an empty space close to which three stars lie, which together with the point of the pole make a rough quadrangle, as Pytheas of Massilia [better known as an explorer] tells us’, trans. Heath 1932 p. 119, who thought the stars in question were probably Draco ̛ and к and UMin β; Roseman 1994 pp. 118-19 prefers nearer stars of lesser magnitude. Eudoxos’ error had been repeated by Euclid, Phainomena Pref. ‘But a certain star is seen between the Bears which does not change from place to place, but turns about the position where it is.’

¹⁹ Jones 1995 p. 49. He does not say by whom they were thus known, but it would appear to be the Phoenicians (see the sources about to be cited in text). Nor is a date given for this label, but the stars would have lined up with the pole around the time of Christ, and would have orbited it tightly for centuries either side of that.

²⁰ This name is surely descriptive of their shape. An explanation for how the Greek name Bear could have derived from a Near Eastern word for wagon is given in Kidd 1997 p. 187, line 27.

²¹ Literally, ‘dog’s tail’: Ursa Minor.

²² Literally, ‘helix’; something with a spiral shape, which goes round, circles, coils. This is the first appearance of this name and surely reflects its circumpolar motion; Homer calls it the Bear. See Kidd 1997 p. 188, line 37.

²³ When mariners had the courage to cross open waters the positions of the sun or the stars were their navigational guides. Deep-water archaeology is re-writing the picture of ancient trade in the Mediterranean, which (as is now apparent) did not always tramp round the coasts in shallow waters but sometimes struck out for a direct crossing.

²⁴ Phainomena 26-44, Mair trans, slightly modified.

²⁵ Think of an earth globe and the tilt of the poles; they point 23.5° off the vertical. Rotate the globe’s stand around 360° and you have this motion. As it moves round, the pole will point to different parts of the ceiling, describing a circle. So in reality earth’s north pole points toward different parts of the sky, describing a cone from the centre of the earth 47° in diameter over the course of nearly 26,000 years.

²⁶ See Plantzos 1997.

²⁷ The range of meteoric phenomena distinguished (e.g. in Aristotle’s Meteorology book 1; ‘burning flames’, ‘shooting stars’, ‘torches’, ‘goats’, ‘chasms’, ‘trenches’, ‘comets’, ‘bearded stars’ and ‘haloes’) illustrates the former point. For the latter, deposits in the polar ice-caps reveal that global atmospheric lead pollution was worse in Roman times than at any other time in history.

²⁸ Archimedes wrote a treatise (lost) On Sphere-making, and Cicero mentions both Archimedes’ and Poseidonios’ own spheres in De Natura Deorum 2.88, Rep 1.21 and 1.28, and Tuse. Disp. 1.63.

²⁹ On such instruments see de Solla Price 1957; also Maddison 1963.

³⁰ On which see de Solla Price 1975.

³¹ e.g. ‘In On the Length of the Year [Hipparkhos] assumes only the motion which takes place about the poles of the ecliptic, although he is still dubious, as he himself declares, both because the observations of the school of Timokharis are not trustworthy, having been made very crudely, and because the difference in time between [Timokharis and himself] is not sufficient to provide a secure result’, Ptolemy Almagest 7.3 Toomer trans. See also n. 4 above.

³² On whom see Gottschalk 1980.

³³ The alternative is of course that the celestial sphere is rotating even faster, but this was acceptable. Indeed, it was even used by Nigidius ‘the potter’ as an argument in defence of astrology to explain why twins can be so different. Augustine City of God 5.3 reports Nigidius’ demonstration of the point using a potter’s wheel.

³⁴ This and the next clause in the argument are based on the notion of natural motion, according to which one would expect light things to move quickly and heavy things to move slowly. Axial rotation theory implies an ‘unnatural’ reversal of this situation, with rapidly moving earth and stationary stars.

³⁵ This paragraph is a paraphrase of Ptolemy’s comments which lead into the section 1.7 quoted above, which I have rewritten to give the modern view of these matters. ‘Certain people’ here means all modern astronomers and astrophysicists. In the original version, Ptolemy presumably meant Herakleides of Pontus, Aristarkhos of Samos, and Seleukos of Seleukia at least, but he omits their names and anonymizes them.

³⁶ ≈30 km/s in Kaufmann 1994 p. 146.

³⁷ ≈230 km/s in Kaufmann 1994 p. 312; see p. 464 for the duration of this revolution.

³⁸ ≈600 km/s in Kaufmann 1994 p. 533.

³⁹ Ekphantos, Hiketas, Aristarkhos and Seleukos sided with Herakleides. In addition, before and after Herakleides the Pythagoreans (who numbered many) believed that the earth did not occupy the middle of the universe, but orbited a central fire; therefore they must have had some answer to the sort of objections raised by Ptolemy.

⁴⁰ See the discussion of this point by Hanson 1973 pp. 18-21.

⁴¹ Specifically, just how far away the stars must be and just how big the universe must be for our ever-changing position not to have any apparent observational effect.

⁴² In the context of a lunar eclipse. Since a lunar eclipse is caused by the earth passing between sun and moon, it should not be possible to observe both sun and eclipsed moon simultaneously; the fact that on occasion both were visible simultaneously suggested to some that light rays were bent by earth’s atmosphere. See Kleomedes, De motu circulari 2.6, conveniently in translation in Heath 1932 pp. 162-6, and discussed by Pederson and Pihl 1974 p. 135.

⁴³ There was no trigonometry in Aristarkhos’ time. With modern trigonometry (specifically the cosine function) it is quick and easy to prove Aristarkhos’ Proposition 7 from his hypotheses. The origins of trigonometry can be found in the chord functions developed and used by Hipparkhos, Menelaos, and later Ptolemy, who drew up tables of their values to enable them to compute astronomical data.

⁴⁴ A quadrant is a quarter of a circle. If we divide it into 360°, a quadrant is then 90°, and one thirtieth ofthat is 3°. Therefore Aristarkhos hypothesizes that the angle is 87°. If you substitute ‘less than a quadrant by one five hundred and fortieth of a quadrant’ (89° 50′) in the rest of the computation, then you’ll get the modern figure, as near as matters.

⁴⁵ A sign of the zodiac is one twelfth of 360°, so 30°. One fifteenth of that is 2°.

⁴⁶ Although Aristarkhos proposed heliocentric theory, he here assumes a geocentric model; the sun A moves round a stationary earth B.

⁴⁷ Aristarkhos has neglected to tell us where to put H. From the demonstration as a whole it becomes clear that it is the point where an extension of BD cuts FE.

⁴⁸ By Pythagoras’ Theorem, for which see chapter 3 (Mathematics).

⁴⁹ Aristarkhos assumes the reader knows this to be true. It can be demonstrated by adding one point, X, to the diagram, and one line, GX. Since G is found by bisecting FBE, GX and GE are equal. FG is the hypotenuse of the triangle FXG, and X is a right angle. Therefore FG is to XG as FB is to FE. Since XG and GE are equal, so FG is to GE as FB is to FE. Note that Aristarkhos is working with squares on the lines, and only reduces to the roots, the lines, in the next step of the argument.

⁵⁰ The square roots of 49 and 25 respectively. He needed to choose numbers with integer square roots in the same proportions as his ratios, and he opted for 49 as not quite (less than) 50, which is double 25.

⁵¹ Additional to Heath’s translation, which has componendo here.

⁵² Aristarkhos now multiplies both by 3, to compare with the 15 as a figure for the quarter of a right angle of 60 parts (done above).

⁵³ Dividing both by 2.

⁵⁴ Mathematikoi usually means more generally ‘learned men’. Intellectual life was not such that learned men pursued one, and only one, subject. Some, however, had greater facility with mathematics than most, and these are clearly the sort of people Lucius has in mind here.

⁵⁵ Taprobane is Sri Lanka. Eratosthenes had located it on the same parallel as Ethiopia.

⁵⁶ It is precisely in the context of Aristarkhos’ heliocentric model that Archimedes advances his ‘Sand-Reckoner’, as a system by which the huge numbers implied by Aristarkhos’ immense cosmos might be expressed.