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Historical astronomy
Ptolemy: on trial for fraud
Ptolemy’s reputation as the greatest astronomer of antiquity has come into question
since the 10th century. Norriss S Hetherington reviews the search for fraud in the Almagest
and highlights other possible explanations for the anomalies that it contains.
F
stars” (Dreyer 1917,
rom AD 150 until
1918). Strengthening the
Copernicus, Ptolemy’s
argument for actual meaAlmagest dominated astsurements by Ptolemy was an
ronomical science. His reported
analysis in 1925 comparing Ptoleobservations, however, match corremy’s and Hipparchus’ data. Discrepsponding theory so closely that the
ancies between the Ptolemaic and HipFrench astronomer Delambre asked: “Did
parchian longitudes were judged to be “too
Ptolemy observe? Are the observations which
great to be explained on the assumption that
he claims to have made not calculations by
Ptolemy simply copied Hipparchus’ catalogue
means of tables, and examples intended for a
with an additional constant” (Vogt 1925; Pedbetter understanding of his theories?” (Delamersen 1974). This conclusion would much later
of stars
bre 1817). Furthermore, a systematic error in
The
title
be disputed (Graßhoff 1990), but it still stood
in HipparPtolemy’s star catalogue led Arab astronomers
page of
unchallenged in the 1970s. Then a geophysichus’ lost catas early as the 10th century to suspect that
Ptolemy’s
cist, Robert R Newton, charged that Ptolemy,
alogue came to
Ptolemy had taken over the results of a pregreat star catala hero to many as the greatest astronomer of
about 850. Ptoldecessor, a suspicion shared by Tycho Brahe in
ogue, the Almagest.
the ancient world, had committed a crime
emy’s catalogue, howthe 16th century, by Lalande in the 18th cenagainst his fellow scientists and scholars,
ever, contains 1022 stars. Dreyer and others
tury, and also by Delambre. Was Ptolemy the
betraying the ethics and integrity of his profesconcluded that more stars must have been
greatest astronomer of antiquity, or the greatsion. The Almagest had “done more damage to
added after Hipparchus, presumably by Ptolest fraud in the history of science?
astronomy than any other work ever written,
emy. The possibility that 850 stars was only a
Ptolemy’s reported longitudes for each star’s
and astronomy would be better off if it had
lower limit did not arise in scholarly arguments
position are too small by about 1°, but his latinever existed. Thus Ptolemy is not the greatest
until much later (Goldstein 1982) and thus did
tudes are basically correct. Precession – a slow
astronomer of antiquity, but he is something
not slow Dreyer’s speculative momentum.
change in the orientation of the Earth’s axis of
still more unusual: he is the most successful
Dreyer next tackled the systematic error in
rotation – changes the longitude of stars over
fraud in the history of science” (Newton
stellar longitude. Ptolemy had measured his
time. Believing that the change in longitude
1977). The crime was all the worse because
longitudes relative to a fundamental star,
was 1° per century, Ptolemy might have added
Ptolemy’s encyclopaedic summary had
which in turn was measured relative to the
22⁄3° to each of Hipparchus’ longitudes for the
replaced much of previous astronomy, thereby
Sun. Since Ptolemy’s solar theory was incorrect
22⁄3 centuries between 129 BC and AD 137,
forever depriving science of fundamental inforin a manner that resulted in too small a solar
the epoch for which Ptolemy’s catalogue was
mation about an important area of astronomy
longitude by about 1°, Dreyer showed, all of
prepared. The actual amount of precession,
and history. The charge of fraud, though not
the star longitudes would be off by the same
however, was 32⁄3°. Hence Ptolemy’s longitudes
new, was worded more emphatically than hithamount. Dreyer now “found no reason to diswould be 1° too small, as in fact they are, on
erto and commanded new attention.
believe his [Ptolemy’s] positive statement, that
average (Evans 1987a,b). J L E Dreyer initially
Newton had been investigating ancient
he had made extensive observations of fixed
accepted this explanation, in 1906 writing that
observations of the
Ptolemy’s star cataMoon’s
position.
logue was “nothing
he astronomical data in Ptolemy’s
explanation, careful examination of the
There is a small accelbut the catalogue of
eration in the motion
Hipparchus brought
great work, the Almagest, has
data suggests that a combination of
of the Moon in its
down to his own time
raised suspicions in the minds of
observational, calculation and
orbit around the
with an erroneous
scholars for some 900 years. Did
rounding errors, plagiarism of existing
Earth and also a slowvalue of the constant
Ptolemy make all or any of the
data and selection of the best
ing down in the
of precession”. ManuEarth’s spin about its
scripts attributing to
observations, did he take them from
examples to support Ptolemy’s new
axis separate from
Hipparchus a Greek
existing catalogues, or are they solely
theories could account for the
that strictly predicted
list of constellations
the results of calculations? Scientific
appearance of fraud. But in the
from gravitational
soon caused Dreyer
fraud is the charge levelled at
absence of categorical evidence, the
theory. The cause is
to re-examine the case
tidal friction between
against Ptolemy. EstiPtolemy. Although this is a possible
verdict must remain “not proven”.
the seas and the solid
mates of the number
T
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April/May 1997 Vol 38 Issue 2
Historical astronomy
Earth, slowing the Earth’s axial spin; conservation of angular momentum in the
Earth–Moon system is conserved as the Moon
speeds up. A comparison of ancient observations with extrapolations from modern theory
offers, potentially, a powerful means of studying the acceleration of the Moon and the retardation of the Earth’s rotation over time (Newton 1970). Newton might have saved himself
much effort and aggravation, however, had he
been aware of earlier studies finding suspiciously close agreement between theory and
observation in the Almagest.
Subsequently, Owen Gingerich found with
regard to Mars that “although Ptolemy’s stated
observations are not particularly good…they
do match his tables in an uncanny way. In fact,
this situation prevails throughout the entire
Almagest, which leads to the suspicion that
something is going on that does not meet the
eye, something craftily concealed in the writing
of the Almagest” (Gingerich 1983a). Also with
Mercury, a close examination finds agreement
between theory and observation too good to be
true (Gingerich 1983b). Ptolemy seemingly
fabricated his stated observations, but the
agreement between his numerical parameters
and modern values is too close to be fortuitous.
the 18 easily recognizable stars that he listed in
the Almagest (VII, 3) with positions recorded
by Timocharis, by Hipparchus and by himself,
he would have found approximately the modern value. He reported, however, precession of
1° in about 100 years, as Hipparchus had
determined earlier. Next Ptolemy showed how
6 of the stars supported this conclusion, without mentioning the other 12. Seemingly, the
fact that a few cases fitted well made them
appear more important, and was sufficient to
provide confidence that the theory was right
cations” were “understood by him as little fibs
allowable in the neatening up of his pedagogy”, not “as lies intended to mislead his
readers about matters he regarded as crucial”
(Wilson 1984). “Here we must remember,
above all, that the Almagest is not like a modern research paper. It is a well-honed textbook, modelled as much as applicable on
Euclid’s Elements. It was no doubt Ptolemy’s
intention to present the procedures with sample data so that future astronomers could see
how it was done and could introduce their own
A new type of science
There are several possible explanations. He
likely had a large number of observations –
either his own or borrowed without acknowledgement, from Hipparchus or from someone
else – and the probable errors largely cancelled
each other out in calculations of parameters
(Britton 1992). Next, Ptolemy might have
selected from among his observations a few in
best quantitative agreement with the theory
and then presented these examples to illustrate
the theory. To the ancient Greeks it may not
have appeared as paradoxical as it now does to
test a theory empirically with a measurement
chosen for its best fit with the theory. Ptolemy
and his contemporaries lacked our modern
understanding of error ranges, standard deviations and the use of mean values from repeated observations – concepts that would have
enabled Ptolemy to propound a theory not necessarily in total and absolute agreement with
every data point obtained, but in less strict
agreement with all of the data points to within
a statistically defined interval around a mean
value. Instead, without any tolerable fluctuation in the agreement between theory and
observation, every measurement would be
understood as an exact result. Consequently,
judicious selection from among many measurements was required (Graßhoff 1990).
Selective use of data by Ptolemy is especially
evident with regard to precession: the movement of the equinoxes relative to the stars. Had
Ptolemy calculated the precession from all of
April/May 1997 Vol 38 Issue 2
This illustration from Gregor Reisch’s “Margarita Philosophica...”, printed in Basel in 1508, shows Ptolemy
crowned and in contemporary dress, using a quadrant to measure the elevation of an astronomical object,
accompanied and encouraged by a woman labelled “Astronomia”. Note the armillary sphere in the corner.
(Pannekoek 1955).
In attempting to demonstrate a new type of
science, converting observational data into
numerical parameters of geometrical models,
Ptolemy may have been led to present “an
‘idealized’ account of the way an astronomer
operates starting from observations” (Toomer
1980). He was “fibbing about what he
observed in order to make the agreement
between theory and phenomena appear perfect
when it was not”. His “fudging and fabri-
observations over a longer temporal baseline in
order to derive even better parameters. For
such a purpose he surely wanted to exhibit
mutually consistent observations, the best he
could find.” Also, “anxious to show how the
model might in principle be most easily justified, and keen not to perpetuate data with
observational errors, he must have also tried in
some way to clean up his reported positions”
(Gingerich 1983a).
Appreciations of Ptolemy’s possible didactic
25
Historical astronomy
Expected distribution of fractional
values in the Almagest
intent and the difficulty of reconciling theory
and observation in ancient times were stimulated by Newton’s angry damnation of the
Almagest. Another important contribution
from Newton is the attention he focused on the
distribution of fractional values in the measurements of stellar positions (Newton 1979),
discussed in the box above.
Nothing in the distribution of fractions in latitude has raised any suspicion among historians or astronomers in search of criminal activity. The distribution in longitude is another
matter. Ptolemy reported 226 stars at 0 fractional degree, 182 at 1⁄6°, only 4 at 1⁄4°, 179 at
1
⁄3°, 88 at 1⁄2°, 246 at 2⁄3° (40 min), no stars at
all at 3⁄4° and 102 stars with a fractional 5⁄6°
measurement. How else can we account for the
peak at 2⁄3°, other than to call it evidence of a
Hipparchian peak at 0 min to which Ptolemy,
to update the catalogue for precession, added
22⁄3°? The other peak, originally at 1⁄2°, would
be shifted to 1⁄6°. Furthermore, if we shift by
2
/3° (40 min) the original 1⁄4° and 3⁄4° categories
(15 and 45 min), they become 55 and 25 min.
However, these slots are not represented by
any of the sixth or quarter fractions. Hence
these occurrences now might be shifted into a
neighbouring slot. Adding the 55 s (original
15 s) to the zero category and the 25 s (original 45 s) to the 20 slot (1⁄3°) produces numbers
nearer to what Ptolemy reported. (Of course,
there is no compelling logic that would have
prevented Ptolemy from rounding the values
down and up instead, or half up and half down
26
250
no. of stars
In the Almagest, Ptolemy reported the latitudes and
longitudes of stars as so many degrees plus fractional parts of a degree, the fractions being 0°, 1⁄6°,
1
⁄4°, 1⁄3°, 1⁄2°, 2⁄3°, 3⁄4° or 5⁄6° (corresponding to 0,
10, 15, 20, 30, 40, 45 and 50 min). This reportorial format suggests that the measurements might
have been made with two different instruments,
one calibrated in sixths of a degree and the other in
quarters (perhaps Ptolemy’s, since there is only one
bright star, Pollux, among these presumably later
additions, and also Ptolemy’s reputation as an
accurate observer falls below that of his predecessor, primarily because Ptolemy had a less accurate
knowledge of the constant of precession than did
Hipparchus). Alternatively, using a single instrument calibrated only in 1⁄2° increments, an observer
could have reported as 1⁄6° those stars falling near
the zero mark, as 1⁄3° those stars near the 1⁄2° mark,
and as 1⁄4° those stars nearly in the centre of the
space between 0 and 1⁄2°, and similarly extrapolated for stars between 1⁄2° and the next higher
degree. The following speculation allows for one
instrument or two.
It is equally likely, with a random distribution of
stars, for stellar coordinates to fall at each equal
interval, say of 5 min. However, with an instrument or instruments calibrated to measure only the
random
1
/6 ° divisions
Ptolemy
300
200
150
100
50
0
0
10
20
30 40
minutes
50
60
The expected effect of using an instrument or
instruments calibrated in sixths or half degrees.
above fractions (0°, 1⁄6°, 1⁄4°, 1⁄3°, 1⁄2°, 2⁄3°, 3⁄4°, and
⁄6°), stars at 5, 25, 35 and 55 min (the 5 min intervals not represented by one of these fractions)
would not be measured as such. Instead, the
observer(s) might round such occurrences up or
down into an adjoining category. Thus any star
with a true coordinate of 5 or 55 min might have a
50:50 chance of finding itself assigned a value of
zero degrees (either 0 or 10 for the non-reported
5 min mark and either 0 or 50 for the non-reported
55 min mark), with the result that we would
expect twice as many stars with a longitude of
5
for each.) The new 1⁄4° and 3⁄4° categories were
produced, so the speculation goes, by shifting
by 40 min the original 35 and 5 min slots,
which were “forbidden levels” with no members. Ptolemy reports only 4 stars with a fractional longitude of 1⁄4° and none at 3⁄4°. The
other original forbidden levels, 25 and 55 min,
are shifted to 5 and 35 min, also forbidden categories, hence also unreported, and not in need
of explanation.
Shifting reference stars
The distribution of fractional longitudes in
Ptolemy’s star catalogue is suggestive of plagiarism. However, Ptolemy might actually have
made his reported measurements against a reference star, later found that his longitude for
the reference star was short by an amount
including a fractional part of 2⁄3° (or too much
by 1⁄3°), and then added this 2⁄3° to (or subtracted 1⁄3° from) all of his genuine measurements.
Alternatively, perhaps he only determined the
absolute longitude of the reference star after
making the relative measurements, and it had a
fractional part of 2⁄3°, which he then added to
each star’s relative longitude. Indeed, three
(Spica, Aldebaran and Antares) of the four
bright stars (Regulus is the other) near the
ecliptic that Ptolemy might conveniently have
used as reference stars have fractional 2⁄3° longitudes in the Almagest. The addition of the
fractional longitude(s) of one or more reference
star, Evans notes, can account for the observed
some number of degrees and no fractional part to
be reported, as otherwise would have been. Similarly, the non-reported 25 and 35 min coordinates,
with a 50:50 chance of being rounded up to 1⁄2° or
down to 1⁄2°, would double the 1⁄2° category. The
10, 20, 40 and 50 min categories (1⁄6°, 1⁄3°, 2⁄3° and
5
/6°) would receive rounded up or down measurements from one category each (from 5, 25, 35 and
55 min), increasing their membership by half.
Hence, if the instrument(s) gave readings only in
sixths and quarters of a degree, we could expect
peaks in the distribution of the reported measurements at 0 and 1⁄2°. Furthermore, we would expect
minimums at 1⁄4° and 3⁄4° (these latter two categories, corresponding to 15 and 45 min, receiving
no rounded up or down additions, because both
adjoining 10 and 20 min categories and 40 and
50 min categories are represented by fractional
measurements).
Quantitatively, starting with, say, 100 stars in
each 5 min interval, we would expect to see reported 100 each at 1⁄4° and 3⁄4°, 150 each at 1⁄6°, 1⁄3°,
2
⁄3° and 5⁄6°, and 200 each at 0 and 1⁄2°. This expectation is realized, in a general way, in the reported
latitudes in Ptolemy’s star catalogue in the
Almagest. There are 236 stars with measurements
of 0 fractional degrees and 198 stars with measurements including 1⁄2°. There are only 88 stars at 1⁄4°
and 50 at 3⁄4°. At 1⁄6°, 1⁄3°, 2⁄3° and 5⁄6° there are
106, 112, 129 and 107 occurrences respectively.
distribution of fractions in longitude without
necessarily denying the genuineness of the
reported observations (Evans 1987b).
More evidence in favour of Ptolemy comes
from further analyses subsequent to those of
Evans, which suggest that the stars were
observed by constellation in distinct observational sequences connected with different
seasons of the year. A constant longitudinal
shift in constellations from Cancer to Sagittarius could be the result of using a specific reference star (probably Spica), while larger differences in the values of systematic errors for the
spring hemisphere hints at the employment of
more reference stars. The fraction 40 min predominates overall but not for the five constellations from Cancer to Scorpio, and for northern but not southern constellations, arguably
ruling out the transformation of data en bloc
from Hipparchus to Ptolemy (Shevchenko
1990; Wlodarcyzk 1990).
Ptolemy’s data were transformed en bloc.
The star catalogue of the Arab astronomer
al-Sûfi, composed in around 964, reports parts
of a degree in latitude in the minute marks corresponding to sixths and quarters of a degree,
with peaks at 0 and 1⁄2. However, longitudes
are reported in minutes as 2, 12, etc (10n + 2),
with the main peak at 22 min. The catalogue is
simply the Almagest with 12°, 42 min added to
all of the longitudes (Evans 1987b). There is
also reason to suspect that the star catalogue of
Ulugh Beg, the grandson of Tamerlane and
ruler of an empire from Samarkand between
April/May 1997 Vol 38 Issue 2
Historical astronomy
1409 and 1449, which reported longitudes
with peaks at 25 and 55 min, was plagiarized
from an earlier one, now lost. However, considerable and detailed information is available
regarding Ulugh Beg’s observatory, not only
from written records but from archaeological
excavation as well, concerning when it was
constructed, who observed there and even
some of the instruments. Furthermore, no earlier star catalogue that could have been the
source for Ulugh Beg’s is known. Suppose that
instead of being borrowed from an earlier
source, most of the reported longitudes actually were observed in around 1443, and then
5 min was subtracted for 6 years of precession
to set the catalogue at the epoch of 1437.
Evans concludes that, “when plausible explanations less drastic than imputations of deception and plagiarism are available, some temperance of judgement is called for” (1987a).
Alexandria or Rhodes?
Another set of data requiring judgement is the
distance of stars from the horizon. Not a single
star listed in Ptolemy’s star catalogue was invisible from Rhodes, the home of Hipparchus and
some 5° north of Alexandria, the purported
observational homebase for the Almagest. An
astronomer in Alexandria might have been
expected to observe stars closer to his extended southern horizon. A detailed statistical
analysis, assuming that ancient astronomers
attempted to observe right down to the horizon, estimates the probability of any individual star being included in the catalogue as a
function of its apparent magnitude, carefully
corrected for atmospheric refraction and
atmospheric extinction. Dennis Rawlins found
a 90% probability that Hipparchus could have
compiled the catalogue, but only 1 chance in
1013 that Ptolemy could have (Rawlins 1982).
The probability number is impressive, but the
premises on which it is based are not beyond
dispute. Other astronomers, including Tycho
Brahe, did not observe all the way down to the
horizon. Also, the frequency function probably
should be higher for stars in constellations and
lower for stars not so singled out for the
observer’s attention, as indeed is the case in
Tycho’s work. Furthermore, the conclusion is
drawn from data on only 11 stars (Evans
1987a).
Another facet of the debate over the relative
likelihood of Rhodes or Alexandria having
been the observing site for Ptolemy’s star catalogue involves atmospheric extinction. This
increases greatly near the horizon, causing a
star near the horizon to appear much dimmer
to an observer at Rhodes than at Alexandria
(with its horizon extended 5° farther south).
Estimating amounts of atmospheric extinction
and comparing the predicted result with modern measurements of magnitude, Evans has
April/May 1997 Vol 38 Issue 2
found that the magnitudes reported by Ptolemy of the six southernmost stars are consistent with having been observed in Alexandria
in Ptolemy’s time, and five of the six stars have
magnitudes inconsistent with having been
observed from Rhodes in Hipparchus’ time.
“On the basis of the magnitudes assigned to
the southernmost stars of the catalogue [Ptolemy’s], it appears more likely that the ancient
observer was located at Alexandria than at
Rhodes” (Evans 1987b).
In yet another recent and ingenious addition
to the debate, Rawlins has analysed the manner of observing with an armillary sphere and
the consequences if the ecliptic of the armillary
sphere is set slightly out of the plane of the true
ecliptic. Given the mean error of about 1° in
longitude in Ptolemy’s observations, whatever
its cause, Ptolemy would have misaligned his
armillary sphere accordingly. This misalignment would be reflected in periodic errors in
the measurements. Indeed, Rawlins insists that
had Ptolemy actually made his observations as
stated, with an armillary, there would be systematic error-waves of half a degree in his
reported latitudes. Yet there are not (Rawlins
1982).
We cannot be sure, however, that our hypothetical measuring procedure corresponds
completely to that of the actual observer.
Indeed, the small periodic error present in
Ptolemy’s latitudes, while not consistent with
observations referred to the equinoctial point,
is not inconsistent with observations referred
to a particular reference star, especially if we
further fiddle with the supposed procedure for
orienting and reorienting the instrument during the process of observing. Even if the
process of observation was the very pure and
simple one described by Ptolemy in the
Almagest, a near infinitude of possible sources
of periodic errors, both in the construction of
the instrument and in its use, forestalls any certain conclusion that the reported body of data
was not observed but calculated (Evans
1987b). Indeed, another scholar concludes
from his study of observing techniques with an
armillary astrolabe that the description of his
observations given by Ptolemy is intrinsically
consistent and could correspond to the real
situation (Wlodarcyzk 1987).
Ptolemy did have an incorrect value for the
latitude of Alexandria, which could have contributed to a periodic error and also suggests
plagiarism. He was off by 15 min of arc, an
error several times as large as one would
expect had Ptolemy actually measured the latitude by the method he described in the
Almagest. Newton concludes that “there is little doubt that he [Ptolemy] simply adopted the
obliquity used by Eratosthenes, with no
attempt to verify it independently” (Newton
1980). An alternative explanation is that Ptole-
my, in calculating the latitude of Alexandria,
observed the zenith distance of the Sun at a
time other than noon and then attempted to
calculate the corresponding correction for the
Sun’s change of position from that time until
noon, but mistakenly added rather than subtracted the “correction” in his overall calculation of the latitude based on the supposed
zenith distance of the Sun at noon (Britton
1969). Ptolemy’s reported observation of an
equinox, in precise agreement with that
extrapolated from an observation by Hipparchus and 30 h later than its actual occurrence, cannot be passed off as an innocent
error (Gingerich 1980).
In recent years, much has been learned
regarding Ptolemy, with new and ingenious
suggestions fitting the known facts frequently
enlivening the debate. For every ostensible conclusion put forward, however, one or more
possible, and too often plausible, alternative
explanations has risen in response. None has
yet proved decisive. Success has been measured
in the indecisive widening and deepening of
our understanding. ●
Norriss S Hetherington is at the Office for the History of Science and Technology, University of California at Berkeley, 543 Stephens Hall #2350, Berkeley, California 94720-2350.
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