On the Eclipse of Hipparchus

Abstract

We investigate the date of observation of the Hipparchus eclipse using our latest measurement of historical variations in the Earth’s rotation to plot the tracks of the potential eclipses. We conclude that Hipparchus most probably analysed the eclipse of −189 in deriving the distance to the Moon, as concluded by Toomer in 1974.

Keywords

Hipparchus eclipses lunar distance

Introduction

As related separately by the early astronomers Pappus and Cleomedes, the great ancient Greek astronomer Hipparchus (ca. −193 to −119) utilised two independent observations of the same solar eclipse to derive the first reasonably accurate value for the lunar distance. The method comprised knowing that an eclipse was total at one place, while at another, close to the same meridian, a fraction of the Sun was visible. A measurement of the fraction uncovered, combined with the knowledge of the size of the arc of the meridian between the two places as a fraction of the Earth’s radius, gives a measure of the lunar distance in units of Earth radii. Hipparchus deduced a distance of 71 Earth radii, which for its time (and for many centuries afterwards) is remarkably close to the correct mean value of 60.

Unfortunately, no record of the date of the eclipse is preserved. Our objective is to deduce the date of the eclipse with the aid of our recent measurement (Stephenson et al.¹) of the rotational phase of the Earth, which is denoted by the parameter $Δ T$ . It is the difference between a uniform timescale (Terrestrial Time) and the time derived from the Earth’s rotation (Universal Time). This allows us to plot the tracks of totality of historical eclipses with unprecedented accuracy. With this tool, we investigate the possible dates for the eclipse.

The times, associated quantities, and locations on the Earth of the umbral shadow have been calculated using our own software based on the Besselian plane method as described in The Explanatory Supplement.^2,3 The basic coordinates of the Sun and Moon are from the Jet Propulsion Laboratory’s long-term ephemerides DE 431 (Folkner et al.⁴) and their apparent places uses the algorithm that is given in The Explanatory Supplement⁵ together with routines from the Standards of Fundamental Astronomy.⁶ The map outlines were downloaded from Natural Earth at naturalearthdata.com.

Biographical details of Hipparchus

Hipparchus was born in Bithynia in NW Asia Minor, probably around −194 (Dicks,⁷ pp. 3ff). He was a citizen of Nicaea and made his earliest observations (of the risings and setting of stars) in or near Nicaea. However, when he was aged about 30 years (ca. −163), he moved to Rhodes, where he seems to have spent the rest of his life. What is probably his earliest known observation at Rhodes (of the autumnal equinox) dates from −162 (Almagest, III, 1; Toomer⁸). He made many further equinox observations down to −127. His last known observations (measurements of the positions of the Sun and Moon) were made in −125 (Almagest, VI, 5). He probably died around −119, although the exact year of his death is unknown.

Records of the eclipse analysed by Hipparchus

Regrettably, most of Hipparchus’ writings survive only in quotations and paraphrases by other authors. However, Pappus, writing in the early fourth century a.d., evidently had direct access to a copy of Hipparchus’ book On Sizes and Distances. In Pappus’ Commentary on the Almagest (V, 11), he relates that in Book I of On Sizes and Distances, Hipparchus utilised in his calculations a solar eclipse which,

in the Hellespontine region was an exact eclipse of the whole Sun, such that no part of it was visible, but at Alexandria by Egypt approximately four-fifths of the diameter was eclipsed.

Pappus adds,

by means of the above he shows in Book 1 that, in units of which the radius of the Earth is one, the least distance of the Moon is 71, and the greatest 83. (trans. Toomer⁹)

Cleomedes¹⁰, writing in the first century a.d., without mentioning Hipparchus, asserts in his De Motu Circularis Corporum (II, 3) that:

Once when the Sun was wholly eclipsed in the Hellespont, it was observed in Alexandria to be eclipsed except for the fifth part of its diameter, which is … except for two digits and a little more.

Cleomedes adds,

Now since it is 5000 stades from Alexandria to Rhodes and just so many from thence to the Hellespont, necessarily a digit will be seen at Rhodes. (trans. Stephenson¹¹)

In his mention of Rhodes, Cleomedes is evidently linking the eclipse with Hipparchus, although he does not directly affirm that Hipparchus investigated the eclipse.

It is significant that neither Pappus nor Cleomedes, while both noting that the Sun was completely eclipsed as seen from the region of the Hellespont, did not give any secondary descriptive details, such as darkness by day or the visibility of stars. Evidently, Hipparchus’ specific concern was that the Sun was completely covered by the Moon. Among ancient Greek and Roman records of solar eclipses, this focus on the complete disappearance of the Sun is quite rare. Other extant ancient European records of large solar eclipses usually only allude to darkness and/or the visibility of stars. In Hipparchus’ determination of the lunar distance, his concern was not the degree of darkness or the visibility of stars. He needed – and obtained – a clear statement that the Sun was wholly obscured at a known location (specifically the Hellespont). There is nothing in these reports to suggest that Hipparchus actually witnessed the eclipse himself, although he could have seen the eclipse of −128 from Rhodes, where he was probably living at the time (Dicks¹²).

The quoted magnitude of 4/5 of the solar diameter at Alexandria by Pappus is unusual; in expressing magnitude, digits (each equal to 1/12 of the solar diameter) were usually preferred by ancient Greek astronomers. Interestingly, Cleomedes gives the magnitude at Alexandria both as a fraction of the solar diameter and in digits. However, his reference to digits may be simply for purposes of explanation. In our subsequent discussion, we shall adopt only the assertion by Pappus (and confirmed by Cleomedes) that as seen from Alexandria approximately 4/5 of the Sun’s diameter was eclipsed.

Hence, in brief, we have two statements that the eclipse was fully total in the Hellespontine region, while it was estimated that 4/5 of the solar diameter was eclipsed in Alexandria.

The Hellespont

The short and narrow Hellespont strait (now known as the Dardanelles) extends in a roughly NE direction linking the Aegean Sea with the Sea of Marmara. It extends between locations with geographic coordinates $40 ° . 0 N$ , $26 ° . 1 E$ to $40 ° . 4 N$ , $26 ° . 7 E$ . However, the term “en tois peri ton Hellesponton topois,” quoted by Pappus in Hipparchus’ discussion of the eclipse, denoted more than just the strait. It specified one of the “climata,” or belts of latitude of width 400 stades or approximately $0 ° . 6$ , but indefinite longitude. In his Commentary on Aratus, Hipparchus uses precisely this expression (“en tois peri ton Hellesponton topois”), asserting that in this location that the longest day (at the summer solstice) is 15 hours. From the Geography of Strabo¹³(II, 5.7) – the first century b.c. author who derived much of his information from Hipparchus – it is possible to fix the latitude limits of the Hellespontine clima with fair accuracy. Towards the south, Alexandria in the Troad ( $39 ° . 8 N$ – not to be confused with Alexandria in Egypt) was included in the Hellespontine clima, but the zone did not reach as far north as Lysimachia $(40 ° . 6 N)$ , a little beyond the east end of the strait. Hence, a latitude of range between about $39 ° . 8 N$ and $40 ° . 4 N$ seems most likely.

On the question of longitude, it is arguable that Hipparchus required for his eclipse calculation a definite location for the Hellespont: he needed both the approximate distance between the Hellespont and Alexandria, and to assume that both places where the eclipse was observed to be approximately in the same meridian. In ancient times, determination of longitude presented considerable difficulties. From Strabo (II, 5.7), Hipparchus, following Eratosthenes, believed that Alexandria $(29 ° . 9 E)$ , Rhodes $(28 ° . 2 E)$ , the Troad $(26 ° . 2 E)$ at the western end of the Hellespont strait, and Lysimachia $(26 ° . 8 E)$ at the eastern end of the strait to be on the same meridian. It thus seems reasonable to infer that the Hellespont eclipse observation was made in or near the strait itself. As Fotheringham¹⁴ noted in his discussion of the eclipse of Hipparchus, both sides of the Hellespont were “studded with Greek cities.” We shall thus assume for our computations a location for the Hellespont eclipse observation within the strait itself, as cited above $40 ° . 0 N$ , $26 ° . 1 E$ to $40 ° . 4 N$ , $26 ° . 7 E$ , with a mean location of latitude $40 ° . 2 N$ , longitude $26 ° . 4 E$ .

Possible dates of the eclipse

There is no indication of the date of the eclipse in the extant records. Limits on the date are set by the foundation of Alexandria (−330) and the death of Hipparchus (ca. −119); his last known observations were made in −125. During this long interval, we compute that only three solar eclipses could be fully total in the vicinity of the Hellespont: −309 August 15, −189 March 14, and −128 November 20. The clear statement by Pappus, evidently based on the Hipparchus’ treatise On Sizes and Distances, that “no part of the Sun was visible” in the Hellespontine region precludes an annular eclipse.

For each of the above three total eclipses, dates, $Δ T$ values, and magnitudes at the Hellespont strait and Alexandria ( $31 ° . 22 N$ , $29 ° . 92 E$ ) are listed in Table 1.

Table 1.

Calculated lunar distance for three eclipses under consideration.

Date	$T (s)$	Greatest eclipse magnitude		Assumed magnitude	Sun’s declination (°)	Calculated distance (Earth radii)
Date	$T (s)$	Hellespont	Alexandria	Assumed magnitude	Sun’s declination (°)	Calculated distance (Earth radii)
$- 309$ August $15$	$14150$	$1.06$	$0.76$	$0.80$	$+ 16$	$86$
$- 189$ March $14$	$12670$	$0.99$	$0.91$	$0.80$	$- 4$	$71$
$- 128$ November $20$	$11960$	$1.01$	$0.80$	$0.80$	$19$	$53$

Toomer calculated the distance of the Moon in Earth radii based, as far as could be ascertained, on the same parameters as those used by Hipparchus. These were as follows:

The eclipse was total in the region of the Hellespont, with an adopted latitude of $41 ° N$ .

For the same eclipse, 4/5 of the Sun’s diameter was covered at Alexandria, with an adopted latitude of $31 ° N$ .

The angular diameters of the Sun and Moon had the same mean value of 0°.55.

The parallax of the Sun was zero.

The Hellespont and Alexandria were on the same meridian.

The trigonometric equivalent of the calculation of the topocentric distance $D^{'}$ in Toomer’s¹⁵ paper (p. 134) is displayed in Figure 1, from which we derive the following relations

D^{'} \sin (Su n^{'} s diameter \times f) = x = 2 \sin \frac{1}{2} (ϕ_{H} - ϕ_{A}) \sin [90 ° - \frac{1}{2} (ϕ_{H} + ϕ_{A}) + δ]

(1)

where $f$ is the fraction of the solar diameter uncovered; $ϕ_{H}$ and $ϕ_{A}$ are the latitudes of the Hellespont and Alexandria, respectively; and $δ$ is the declination of the Sun. Inserting the values of the parameters and adding 1 for the geocentric distance $D$ , we find

D \approx 10 \cos (36 ° - δ) / 0.11 + 1 Earth radii

Figure 1.

Diagram showing the relationship between Alexandria (A), the Hellespont (H), and the parallactic displacement of the Moon.

The results of the calculation for the Moon’s distance for each eclipse, with an assumed magnitude of $0.80$ at Alexandria, are listed in Table 1. The calculation is sensitive to the Sun’s declination in each eclipse, and the results are differentiated largely by the variation in declination, which, fortuitously, differs appreciably between them. All the other parameters were assumed to be the same for each eclipse: i.e., the eclipse was total at the Hellespont and 4/5 of the Sun’s diameter was covered at Alexandria. A reduction in the adopted latitude of the Hellespont by $1 °$ to $40 ° N$ decreases the results for the Moon’s distance by about 10%.

For eclipse −189, the computed distance of 71 Earth radii agrees with Hipparchus’ value, and on that basis, Toomer concluded that this was indeed the eclipse Hipparchus used.

We now turn our attention to the tracks of totality for these three eclipses, using values of $Δ T$ taken from the cubic spline in our paper,¹⁶ which was fitted to the best historical data considered by us then. The tracks of totality for these eclipses are plotted in Figures 2 to 4. We discuss these in turn.

Figure 2.

Track of totality on −309 August 15.

Figure 3.

Track of totality on −189 March 14.

Figure 4.

Track of totality on −128 November 20.

−309 August 15

The earliest of the three eclipses in Table 1 occurred not long after the foundation of Alexandria. Figure 2 shows that this eclipse was total at the Hellespont, and the diameter of the Sun covered at maximum eclipse at Alexandria was 3/4 (0.76), as distinct from the reported value of 4/5 (0.80), which Hipparchus used in his estimation of the lunar distance. These are tolerably close, and hence, this eclipse satisfies the two observational conditions.

Toomer rejected −309 on the basis of the calculated large value (86 Earth radii) for the lunar distance (see Table 1). Besides this, there are other mitigating circumstances that argue against this eclipse. The eclipse of −309 occurred several generations before Hipparchus’ own time. For his calculation of the lunar distance, Hipparchus needed to be able to confidently synchronise two reliable and accurately dated independent records of the same eclipse from the Hellespont and Alexandria – two widely separated sites. This might well have presented him with severe problems. As Stephenson¹⁷ (pp. 357–8) noted, only two other ancient writers are known to have obtained access to independent observations of the same eclipse from long before their own time. Uniquely, both Pliny of Rome (writing in the first century a.d.) and Ptolemy (second century a.d.) cited two brief independent accounts of the lunar eclipse of −330 September 20.

In his investigation, Hipparchus felt able to put sufficient trust in the two selected observations to use them in a determination of the lunar distance. In particular, judging from the generally poor quality of extant Greek records of large solar eclipses, Hipparchus would have had little justification for assuming that in −309, many years before his own time, the whole Sun was indeed exactly obscured at the Hellespont, with no part of it visible. He also had to be sure of the accuracy of the fraction covered at Alexandria. For these reasons, we eliminate −309 as a contender.

−189 March 14

This eclipse (see Figure 3) occurred when Hipparchus was a child (perhaps only about 4 years old), living in Nicaea. The eclipse was total there and his family could have recollected witnessing it. This could have helped him establish a reliable date for the eclipse.

Toomer remarked that, as noted by Ptolemy (Almagest, IV and V), the year $- 189$ fell in the middle of a period from which Hipparchus obtained four reports of lunar eclipses (but no solar events) from Alexandria (−200 to −173). Toomer suggested that it was possible that the same person made all four observations – as well as reporting the magnitude of the solar eclipse of −189. However, this notion – although possible – is again no more than speculation.

The plot of the track in Figure 3, computed with the value of $Δ T$ from our paper¹⁸, shows that the track of totality passed slightly to the south-east of the strait. Although the observed fraction of the Sun’s diameter covered at Alexandria was estimated to be 8/10, calculation shows that a little more than 9/10 was actually covered at maximum phase.

Further, as is evident from Figure 3, we compute that the zone of totality would have extended from near the Hellespont a considerable distance down the west coast of Asia Minor. It might well be asked why Hipparchus would select the Hellespont on this occasion when there were many other widely separated places over a range of as much as 400 km to the south where the eclipse would have indeed been total. However, the larger arc subtended by the more distant Hellespont, would have provided greater resolution of the lunar distance.

In summary, the eclipse of −189 fails on two counts: it was apparently not total at the Hellespont and the reported magnitude at Alexandria is significantly different from the calculated greatest magnitude. Nevertheless, according to Toomer, the computed lunar distance agrees with Hipparchus’ result. We will return to these points after the consideration of −128.

−128 November 20

This eclipse (see Figure 4) occurred late in Hipparchus’ life, only about 3 years before his last observation and perhaps about a decade before his death. However, only in the case of this eclipse could Hipparchus have obtained the reliable observations which he needed to derive a viable result for the lunar distance. As shown in Figure 4, our computed track with a value of 11960 seconds for $Δ T$ completely covered the Hellespont strait. The computed maximum fraction of the Sun’s diameter covered at Alexandria of 0.80 (Table 1) agrees exactly with the value reported to have been used by Hipparchus (0.80). The computed altitude of the Sun at Alexandria was $11 °$ . At this comparatively low altitude, the glare of the crescent Sun would have been reduced and made direct observation of the fraction of the Sun covered easier to estimate than in −189.

So, this eclipse presents a prima facie case: totality at the Hellespont and 4/5 covered at Alexandria. However, Toomer’s computed lunar distance of 53 Earth radii is at variance with Hipparchus’ value (71). We now compare and contrast the relative merits of −189 and −128.

−189 versus −128

Both of these eclipses have drawbacks when trying to settle, which is the Hipparchus eclipse. Eclipse −189 has two problems: according to our calculations, it was not total at the Hellespont, and the magnitude of the eclipse at Alexandria was too great. Can we resolve these two issues?

First, our calculated position of the track is directly dependent on the value of $Δ T$ . A reduction of 300 seconds in the value of $Δ T$ around the epoch −189 would move the longitude of the track 1°.25 west, just bringing the belt of totality over the Hellespont. The value of $Δ T$ in our spline fit is heavily dependent on the value of $Δ T$ from the very reliable eclipse of −135 observed at Babylon. The possible range of $Δ T$ from that observation is 11220–12140 seconds. Our spline value in our paper¹⁹ for −135 is 12070 seconds. A revised value 300 seconds less than this (11770 seconds) is still comfortably within the prescribed limits. So, the range of uncertainty in our values of $Δ T$ around the epoch −189 does not preclude the possibility that this eclipse was total at the Hellespont.

The second drawback of −189 is the difference between the reported value of (2/10) and the fraction (1/10) of the Sun uncovered at maximum magnitude (or phase) at Alexandria. As the deduced lunar distance is inversely proportional to the observed phase, Hipparchus’ resultant value of the distance would have been increased by a factor of 2 if he had used the fraction 1/10 uncovered at maximum phase.

However, the maximum phase is not what is required in computing the lunar distance from equation (1). Ideally, Hipparchus needed the phase at Alexandria at the time when the shadow cone crossed the meridian through Alexandria. This value is 0.75; i.e., 1/4 of the Sun’s diameter was uncovered. With this value, he would have obtained a distance of 57 Earth radii, which is close to the correct value. From Figure 3, we see that the track of totality turned steeply to the north, and, as a result, the phase at Alexandria decreased from a maximum of 0.91 to 0.75 in about 18 minutes. Of course, he could not have been in possession of such detailed information for an eclipse which occurred when he was about 4 years old.

The fact is that according to the report, he used a phase of 0.80, corresponding to a fraction of 1/5 of the Sun’s diameter uncovered. We suppose that the recorded phase of 0.80, corresponding to a fraction of 1/5 of the Sun’s diameter uncovered, was an estimate handed down over a period of many years. This lies in the range of 0.91–0.75 and is tolerably accurate for the second century b.c. Consequently, this second objection to −189 is mitigated somewhat.

Cleomedes’ statement that “Now since it is 5000 stades from Alexandria to Rhodes and just so many from thence to the Hellespont, necessarily a digit will be seen at Rhodes.” cannot apply to −189 because the wide path of the eclipse (Figure 3) precludes the application of an approximately equal ratio of the distances between the Hellespont–Rhodes–Alexandria. Although, his surmise (if it is indeed thus) would be valid in −128! More significantly, why did Cleomedes mention Rhodes at all? Was it because (writing in the first century a.d.) he knew Hipparchus was living there at the time of the eclipse?

From a pragmatic point of view, obtaining reliable observations of the eclipse in −128 had potentially several advantages over −189. It occurred in Hipparchus’ active lifetime, and he would have been able to contact reliable observers at the Hellespont and Alexandria. He was probably resident in Rhodes at that time, and he could have witnessed that about 9/10 of the Sun was covered at maximum phase, which corresponded well with the report of 8/10 from Alexandria. Regarding the measurement in Alexandria, the low altitude of the eclipsed Sun $(11 °)$ would have reduced the glare of the crescent, making the estimation of the uncovered fraction surer. It is unlikely that such firsthand knowledge of the eclipse in −189 would have been available to him. In −128, he had the possibility of validating and co-ordinating the observations.

The main stumbling block with −128 is why Hipparchus obtained 71 Earth radii rather than our calculated distance of 53 using ostensibly the same values of the parameters. We consider three possible reasons why he did not get closer to the correct distance of 57 Earth radii for −128:

Hipparchus had to assume that the Hellespont lay on – or close to – the meridian of Alexandria and that the observed phase at Alexandria corresponded to the passage of the eclipse across its meridian. In fact, the track of totality crossed the meridian of Alexandria at a latitude of about $41 ° N$ (see Figure 4), which is about 1° higher than that of the Hellespont. However, in the calculation of the lunar distance, we adopted Hipparchus’ value of $41 ° N$ for the latitude of the Hellespont. Therefore, the adopted difference of 10° between the Hellespont and Alexandria in Hipparchus’ calculation is about correct for the meridian of Alexandria, and this cannot be a significant source of error in the calculated lunar distance.

Could plausible errors in Hipparchus’ values of the Sun’s declination or its diameter have contributed to the disparity? An error of $\pm 1 °$ in declination would change the result by about $\pm 1$ Earth radii. If he had adopted a diameter of $\frac{1}{2} °$ , he would have obtained 59. These are too small.

Did he neglect the declination altogether? Without the declination, we obtain a distance of 74 Earth radii from equation (1), which is closer to his value. It seems unlikely that he would have ignored the declination.

Conclusion

Either one of the eclipses of −189 or −128 could have been used by Hipparchus to measure the distance of the Moon. From our plotted tracks of totality, we find that the eclipse of −128 satisfies the two observational constraints: total at the Hellespont and 4/5 of the Sun’s diameter covered at Alexandria. Ostensibly, the eclipse of −189 satisfies neither. But these objections to −189 can be circumvented by adjusting the value of $Δ T$ and accepting that the reported fraction uncovered of 1/5 at Alexandria is feasible.

From practical considerations of time and place, the eclipse of −128 seems more favourable. It occurred in Hipparchus’ active lifetime, when he would have been in a position to question reliable observers and cross-compare their reports with his own possible experience of the eclipse at Rhodes. However, the resultant distance using −128 is 53 Earth radii, which is at variance with Hipparchus’ value of 71.

Removing the practical obstacles to the candidature of −189 leaves intact Toomer’s conclusion from a purely mathematical analysis.

Was it cloudy late in the afternoon of −128 November 20, thus denying Hipparchus this seemingly more promising option?

Footnotes

Acknowledgements

The authors acknowledge HM Nautical Almanac Office and the International Astronomical Union’s Standards of Fundamental Astronomy.

Notes on contributors

Leslie V. Morrison worked in the fields of astrometry and the Earth’s rotation at the Royal Greenwich Observatory, 1960–1998. He has collaborated with F. Richard Stephenson on several studies of the long-term variations in the Earth’s rotation, including, Measurement of the Earth’s rotation: 720 b.c. to a.d. 2015.²⁰

F. Richard Stephenson is an Emeritus Professor at the University of Durham, in the Physics department. His research concentrates on historical aspects of astronomy, in particular, analysing ancient astronomical records to reconstruct the history of Earth’s rotation, as exemplified in his book Historical Eclipses and Earth’s Rotation.²¹

Catherine Y. Hohenkerk worked at the Royal Greenwich Observatory from 1971 and in the late 1970s transferred to HM Nautical Almanac Office (HMNAO), where she worked on the Nautical and Astronomical Almanacs. She retired from the UK Hydrographic Office, latterly HMNAO’s host organisation, at the end of January 2017. She is a fellow of the Royal Institute of Navigation and chair of the International Astronomical Union’s Standards of Fundamental Astronomy (SOFA) Board. She collaborated with Stephenson and Morrison in the paper Measurement of the Earth’s rotation: 720 b.c. to a.d. 2015.²²