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all of which are lost except for fragments quoted in other books, especially by Ptolemy in the Almagest. What we have is nevertheless astonishing; they convincingly demonstrate his prowess as an observer and a modeler. For example, in the third treatise, Hipparchos examined the intercalation problem, further improving on the solution of Kallippos (itself an improvement over Meton—see §4.2). Kallippos took an exact value of 3651/4 (and Meton and an astronomer named Euktemon had used 36574 + 1/76 d) for the length of the tropical year, but Hipparchos found a more accurate length (1/300 of a day shorter or 365d2467 compared with the modern 365d2422). Comparing his observed (fractional) date of a summer solstice in 135 b.c. with that made by Aristarchos in 280 b.c., Hipparchos is quoted as writing (in the second treatise listed above) that

It is clear, then, that over 145 years, the solstice occurs sooner than it would have with a [365]1/4-day year by half the sum of the length of day and night. .. . (Almagest; III, 1, H207; in Toomer 1984, p. 139)

(yielding 365d2466) and in the third treatise, is quoted as saying,

As for us, we find the number of whole months comprised in 19 years to be the same as they [Meton, Euktemon, Kallippos] but we find the year to be even less than 1/4[-day beyond 365], by approximately 1/300 of a day. Thus in 300 years its accumulated deficit is 5 days compared with Meton, and 1 day compared with Kallippos'. (Almagest; III, 1, H207; in Toomer 1984, p. 139)

Hipparchos's study of the length of the year led to the realization that the length of the year—i.e., the time for the Sun to complete an annual circuit of the ecliptic—depended on whether it was measured with respect to the stars or to the equinoxes and solstices.

One of his most famous works was the compilation of a star catalog, the first of which we are aware. It contained 850 objects and gave both the position on the sphere and a measure of their brightnesses: magnitudes. In comparing previously measured positions of stars by Aristyllos and Timocharis (who lived in the first quarter of the 3rd century b.c.) with his measured positions, Hipparchos, we are told by Ptolemy,40 discovered the precession of the equinoxes (see §3.1.6 for a detailed discussion of precession):

The ancients were in disagreement and confusion in their pronouncements on this topic, as can be seen from their treatises, especially those of Hipparchos, who was both industrious and a lover of truth. The main cause of the confusion on this topic which even he displayed is the fact that, when one examines the apparent returns . . . to .. . equinox or solstice, one finds that the length of the year exceeds 365 days by less than 1/4-day, but when one examines its return to .. . the fixed stars it is greater . . . Hence Hipparchos comes to the idea that the sphere of the fixed stars too has a very slow motion, which just like that of the planets, is towards the rear. . . . (Toomer 1984, p. 131)

His value for the precession was ~45 arc-sec/year (compared with the modern value, 50.29 arc-sec/year) and was more accurate than was Ptolemy's asserted value, 36 arc-sec/year. It has been argued (see citations given by Sarton 1952/1970, I, p. 445) that it was possible that Hipparchos

40 Almagest, Book VII (Toomer 1984, p. 321).

made use of very old Mesopotamian data. However, Ptolemy (Almagest, Toomer 1984, p. 321) says that Hipparchos had found very few observations of fixed stars before his own time, in fact practically none beside those recorded by Aristyllos and Timocharis, and even these were neither free from uncertainty nor carefully worked out.. . .

Hipparchos is credited also with working out the inequality of the seasons, a result based on precise measurements of the dates of the equinoxes and of the solstices. This discovery led him to establish a theory for the Sun's motion based on the notion of an eccentric, a circle whose center is offset from the Earth. He found the size of the offset to be 1/24 of the radius of the orbit. This formulation accounts for the apparently faster motion of the Sun on the ecliptic during one-half of the year (when the Earth is nearer perihelion) and slower at the other (when the Earth is nearer aphelion), and because an offset circle is a close numerical approximation to an ellipse of very small eccentricity, such a theory is suitably precise for predictions of solar motion to the limited precision of naked-eye astronomy before Tycho Brahe. Hipparchos studied the Moon as well, improving on Aristarchos's values for the Moon's distance,41 arriving at 29.5 Earth diameters (compared with the modern value, 30.1). Finally, his theories of the Moon and Sun permitted the calculation of eclipses to a precision not previously possible; and the determination of the effect of lunar parallax on eclipses made possible a prediction for the location for the eclipse to be visible.

The last great astronomer of classical antiquity was Claudius Ptolemaeus, better known to us as Ptolemy [100-175 a.d.]. He worked in Alexandria, the most prominent astronomical center at the time. His important astronomical writings, maOhmaxtKq cuvxaXtZ (A Systematic Mathematical Treatise), became the standard text for more than a millennium, an endurance record exceeded only by Euclid's Elements, among scientific books. By the 16th century, it was found in numerous Greek, Arabic, and Latin versions under its Arabic title, the Al Magest. Toomer's (1984) English translation is accompanied by copious and informative notes, and it is indispensable for anyone studying the astronomy of Hellenistic Greece. Toomer (1984, p.

5) believes that a considerable number of erroneous interpolations, placed there in antiquity, are found in all existing versions. The work that has been handed down to us now consists of 13 books, possibly organized this way long after Ptolemy's death. Table 7.14 briefly summarizes the content of each of the books.

The Almagest was not intended as a primer, but for those who had "made some progress in the field" (Toomer 1984, p.

6) and were acquainted with the spherical trigonometry in the writings of Autolycus, Euclid's Phaenomena, and Theo-dosius's Sphaerica. The work contains many observations of

41 Hipparchos made use of a solar eclipse that was total at the Hellespont, but partial at Alexandria, for example, to compute the distance to the Moon. The maximum angular extent of the uneclipsed part of the Moon is the angle at the Moon between the two sites at Earth; the linear distance between the sites was known, and the distance to the Moon follows.

Table 7.14. Brief summary of contents of the Almagest.

Book

Nature of the material

I The Earth and its relations to the heavens

II The sky as seen from different places on Earth

III Solar motion;the year

IV Lunar phenomena and motions

V The astrolabe;the lunar orbit and the distances and sizes of the Earth, Sun, and Moon

VI Solar and lunar conjunctions and eclipses

VII The fixed stars;precession;northern hemisphere constellations

VIII southern hemisphere constellations;rising and setting phenomena

IX Planetary orbits;Mercury

X Venus and Mars

XI Jupiter and Saturn

### XII Retrograde motions and greatest elongations

XIII Planetary latitudes and orbital inclinations the Moon and other objects both by Ptolemy and by others, including Hipparchos, Timocharis, Menelaus, and Agrippa, descriptions of phenomena and of instrumentation needed to make careful positional measurements, and a number of hypotheses regarding planetary motion and the layout of the cosmos. He presents the geocentric theory with eccentrics and epicycles to satisfy the observations. It is from this work that the geocentric solar system received its canonical form, and against which post-Renaissance and Reformation European science had to struggle. In his view, the planets Mercury and Venus revolved epicyclically, around a (mean) center that was either on—or very close to—the Earth-Sun line. The Sun, of course, revolved about the Earth and, therefore, so did the inferior planets. However, planetary motion suggested that the deferent orbits were eccentric, i.e., not centered on Earth, which meant that the planets are sometimes closer to, and sometimes farther from, the Earth. Ptolemy determined the apogee and perigee locations of the inferior planets by observing the maximum elongations of these planets from the Sun and noting when the elongations reached extremes. A large maximum elongation angle was assumed to arise because of a smaller distance from the Earth. The geometry is illustrated in Figure 7.16, adapted from Toomer (1984, Fig. 9.5, p. 454). Because Earth and Venus are in resonance, Ptolemy was not able to explore the full extent of Venus's orbital geometry, and the results are not as good as for Mercury.

Among Ptolemy's major improvements were an accurate distance of the Moon, a better lunar theory, and a new star catalog. The lunar distance determination involves parallax, measured, naturally enough, with a "parallactic instrument" (see Figure 3.20; Toomer 1984, Fig. G, p. 245), which was used to measure the meridian zenith distance of the Moon at each solstice. First, Ptolemy observed the range of extremes of the Moon's celestial latitude: ±5°0'. With the symmetry of the Moon's declination variation assumed, the difference of the Moon's extreme zenith distances could be compared, and the departure from symmetry attributed to parallax. At summer solstice, Ptolemy writes, the Moon appeared very nearly in the zenith (z = 21/8° for the Moon's center) as viewed from Alexandria, and at winter solstice, the Moon appeared at its greatest zenith distances. Comparing its true and apparent zenith distances, Ptolemy determined the lunar parallax to be 1°7' = 1°12 when the Moon was at z = 49°48', where the refraction correction is less than 1' (see §3.1.3). This can be compared with the modern value for the lunar mean equatorial horizontal parallax: 0°951, a difference of less than 20%. Ptolemy was also aware that the Moon's distance varied with its position in the orbit. Ptolemy presents a table of lunar parallaxes, however, that are rather divergent, varying between a minimum of 53'34" to a maximum of 63'51" at full and new moons, when the moon is at the highest and lowest points of the epicycle, respectively, but also as high as 1°19' and 1°44' at the quarters; presumably, these are derived from the theory rather than observationally determined. It must be remembered that both Hipparchos and Ptolemy were attempting to describe the motions of Moon and Sun by differences in angular rates due to distances, alone, assuming constant orbital speeds (see §2.3.5). Ptolemy also measured the angular diameters of the Sun and Moon, but failed to find any variation in the size of the Sun over the course of the year.

His orbital theory for the Moon marked a substantial improvement over Hipparchos's. Gingerich (1993, pp. 59-62) summarizes Ptolemy's approach very nicely by illustrating the errors in lunar longitude against lunar phase. In the first instance, a theory that Ptolemy attributes to Hip-parchos features a deferent centered on Earth + an epicycle + an advance of the perigee. It fits the data well at full and new moons and nowhere else, with largest errors ~±3° at the quarter phases. In the second instance, Ptolemy's correction to this model involves an eccentric, the center of which is permitted to rotate. The effect, known as the evection, was thus first reported by Ptolemy. This reduced the longitude errors in the orbit by more than 1/3. The largest residuals now appeared at the octants of the orbit (a further improve-

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