I.
INTRODUCTION
1. In the following pages an attempt is made to offer a survey of the
present state of the history of ancient astronomy by pointing out relationships
with various other problems in the history of ancient civilization and
particularly by enumerating problems for further research which merit our
interest not only because they constitute gaps in our knowledge of an cient
astronomy but because they must be clarified in order to lay a solid foundation
for the understanding of later periods.
I wish to emphasize from the very beginning that the attitude taken here
is of a very personal character. I do not believe that there is any single
approach to the history of science which could not be replaced by very
different methods of attack; only trivialities permit but one interpretation. I
must confess still more: I cannot even pretend to be complete in the selection
of topics essential for our understanding of ancient astronomy, nor do I wish
to conceal the fact that many of the steps which I myself have taken were dictated
by mere accident. To mention only one example: without having been brought into
contact with a recently purchased collection of Demotic papyri in Copenhagen, I
would never have undertaken the investigation of certain periods of Hellenistic
and Egyptian astronomy which now seem to me to constitute a very essential link
between ancient and medieval astronomy. In other words, though I have always
tried to subordinate any particular research problem to a wider program of
systematic analysis, the impossibility of elaborate longrange planning has
again and again been impressed upon me. The situation is comparable to entering
a vast mountainous region on a single trail; one must simply follow the winding
path, trying to give account of its general direction, but one can never
predict with certainty what new vistas will be exposed at the next turn.
2. The enormous complexity of the study of ancient astronomy becomes
evdent if we try to make the first, and apparently simplest, step of
classification: to distinguish between, say, Mesopotamian, Egyptian, and Greek
astronomy, not to mention their direct successors, such as Hindu, Arabic, and
medieval astronomy. Neither geographically nor chronologically nor according to
language can clear distinctions be made. Entirely different conditions underlie
the astronomy in Egypt of the Middle and New kingdoms than in the periods after
the Persian conquest. Greek astronomy of Euclid's time has very little in
common with Hipparchus' astronomy only a hundred and fifty years later. It is
evident that it is of very little value to speak about a "Babylonian"
astronomy regardless of period, origin, and scope. And, worst of all, the
concept "astronomy" itself undergoes changes in meaning when we speak
about different periods. The fanciful combination of a group of brilliant stars
to form the picture of a "bull's leg" and the computation of the
irregularities in the moon's movement in order to predict accurately the
magnitude of an eclipse are usually covered by the same name! For
methodological reasons it is obvious that a drastic restriction in terminology
must be made. We shall here call "astronomy" only those parts of
human interest in celestial phenomena which are amenable to
mathematicaltreatment. Cosmogony, mythology, and applications to astrology must
be distinguished as clearly separated problems —not in order to be disregarded
but to make possible the study of the mutual influence of essentially different
streams of development. On the other hand, it is necessary to coordinate
intimately the study of ancient mathematics and astronomy because the progress
of astronomy depends entirely on the mathematical tools available. This is in
conformity with the concept of the ancients themselves: one need only refer to
the original title of Ptolemy's "Almagest," namely,
"Mathematical Composition."
3. The study of ancient astronomy will always have its center of gravity
in the investigation of the HellenisticRoman period, represented by the names
of Hipparchus and Ptolemy. From this center three main lines of research
naturally emerge: the investigation of the previous achievements of the Near
East; the investigation of preArabic Hindu astronomy; and the study of the
astronomy of late antiquity in its relation to Arabic and medieval astronomy.
This lastmentioned extension of our program beyond antiquity proper is not
only the natural continuation of the original problem but constitutes an
integral part of the general approach outlined here. Astronomy is the only
branch of the ancient sciences which survived almost intact after the collapse
of the Roman Empire. Of course, the level of astronomical studies dropped
within the boundaries of the remnants of the Roman Empire, but the tradition of
astronomical theory and practice was never completely lost. On the contrary,
the rather clumsy methods of Greek trigonometry were improved by Hindu and
Arabic astronomers, new observations were constantly compared with Ptolemy's
results, etc. This must be paralleled with the total loss of understanding of
the higher branches of Greek mathematics before one realizes that astronomy is
the most direct link connecting the modern sciences with the ancient. In fact,
the work of Copernicus, Brahe, and Kepler can be understood only by constant
reference to ancient methods and concepts, whereas, for example, the meaning of
the Greek theory of irrational magnitudes or Archimedes' integrations were
understood only after being independently rediscovered in modern times.
There are, of course, very good reasons for the fact that ancient
astronomy extended with an unbroken tradition deep into modern times. The
structure of our planetary system is such that it is simple enough to permit
the achievement of relatively farreaching results with relatively simple mathematical
methods, but complicated enough to invite constant improvement of the theory.
It was thus possible to continue successfully the "ancient" methods
in astronomy at a time when Greek mathematics had long reached a dead end in
the enormous complication of geometric representation of essentially algebraic
problems. The creation of the modern methods of mathematics, on the other hand,
is again most closely related to astronomy, which urgently required the
development of more powerful new tools in order to exploit the vast
possibilities which were opened by Newton's explanation of the movement of the
celestial bodies by means of general principles of physics. The confidence of
the great scientists of the modern era in the sufficiency of mathematics for the
explanation of nature was largely based on the overwhelming successes of
celestial mechanics. Essentially the same held for scholars in classical times.
In antiquity, mathematical tools were not available to explain any physical
phenomena of higher complexity than the planetary movement. Astronomy thus
became the only field of ancient science where indisputable certainty could be
reached. This feeling of the superiority of mathematical astronomy is best
expressed in the following sentences from the introduction to the Almagest:
"While the two types of theory could better be called conjecture than
certain knowledge —theology because of the total invisibility and remoteness of
its object, physics because of the instability and uncertainty of matter— . . .
. mathematics alone . . . . will offer reliable and certain knowledge because
the proof follows the indisputable ways of arithmetic and geometry."'
II. EGYPT
4. A few words must be said about Egyptian mathematics before discussing
the astronomical material. Our main source for Egyptian mathematics consists of
two papyri—certainly not too great an amount in view of the length of the
period in question! Still, it seems to be a fair assumption that we are well
enough informed about Egyptian mathematics. Not only are both papyri of very
much the same type but all additional fragments which we possess match the same
picturea picture which is paralleled by economic documents in which occur
precisely those problems and methods which we find in the mathematical papyri.
The Egyptian mathematical texts, furthermore, find their direct continuation in
Greek papyri, which again show the same pattern. It is therefore safe to say
that Egyptian mathematics never rose above a very primitive level. So far as
astronomy is concerned, numerical methods are of primary importance, and,
fortunately enough, this is the very part of Egyptian mathematics about which
we are best informed. Egyptian arithmetic can be characterized as being
predominantly of an "additive" character, that is, its main tendency
is to reduce all operations to repeated additions. And, because the process of
division is very poorly adaptable to such procedures, we can say that Egyptian
mathematics does not provide the most essential tools for astronomical
computation. It is therefore not surprising that none of our Egyptian
astronomical documents requires anything more than simple operations with
integers. Where the complexity of the phenomena exceeded the capacity of
Egyptian mathematics, the strongest simplifications were adopted, consequently
leading to little more than qualitative results.
5. The astronomical documents of purely Egyptian origin are the
following: Astronomical representations and inscriptions on ceilings of the New
Kingdom, supplemented by the socalled "diagonal calendars" on coffin
lids of the Middle Kingdom and by the DemoticHieratic papyrus "Carlsberg
1." Secondly, the Demotic papyrus "Carlsberg 9," which shows the
method of determining new moons. Though written in Roman times (after A.D.
144), this text undoubtedly refers to much older periods and is uninfluenced by
Hellenistic methods. A third group of documents, again written in Demotic,
concerns the positions of the planets. In this case, however, it seems to be
very doubtful whether these tables are of Egyptian origin rather than products
of the Hellenistic culture; we therefore postpone a discussion to the section
on Hellenistic astronomy. The last group of texts is again inscribed on
ceilings and has been frequently discussed because of their representation of
the zodiac. There can be no doubt that these latter texts were deeply
influenced by nonEgyptian concepts characteristic for the Hellenistic period.
The same holds, of course, for the few Coptic astronomical documents we
possess. It is, finally, worth mentioning that not a single report of
observations is preserved, in strong contrast to the abundance of observational
records from Mesopotamia. It is hard to say whether this reflects a significant
historical fact or merely that we are at the mercy of the accidents of
excavation.
Speaking of negative evidence, three instances must be mentioned which
play a more or less prominent role in literature on the subject and have
contributed much to a rather distorted picture of Egyptian astronomy. The first
point consists in the idea that the earliest Egyptian calendar, based on the
heliacal rising of Sothis, reveals the existence of astronomical activity in
the fourth millennium B.C. It can be shown, however, that this theory is based
on tacit assumptions which are very implausible in themselves and that the
whole Egyptian calendar does not presuppose any systematic astronomy
whatsoever. The second remark concerns the hypothesis of early Babylonian
influence on Egyptian astronomical concepts. This theory is based on a
comparative method which assumes direct influence behind every parallelism or
vague mythological analogy. Every concrete detail of Babylonian and Egyptian
astronomy which I know contradicts this hypothesis. Nothing in the texts of the
Middle and New Kingdom equals in level, general type, or detail the
contemporaneous Mesopotamian texts. The main source of trouble is, as usual,
the retrojection into earlier periods of a situation which undoubtedly
prevailed during the latest phase of Egyptian history. This brings us to the
third point to be mentioned here: the assumption of an original Egyptian
astrology. First of all, there is no proof in general for the widely accepted
assertion that astrology preceded astronomy. But especially in Egypt is there
no trace of astrological ideas in the enormous mythological literature which we
possess for all periods.
The earliest horoscope from Egyptian soil, written in Demotic, refers to
A.D. 13; the earliest Greek horoscope from Egypt concerns the year 4 B.C. We
shall presently see that the assumption of a very late introduction of
astrological ideas into Egypt corresponds to various other facts.
6. It is much easier to show that certain familiar ideas about the
origin of astronomy are historically untenable than to give an adequate survey
of our real knowledge of Egyptian astronomy. A. Pogo is to be credited with the
recognition of the astronomical importance of inscriptions on the lids of a
group of coffins from the end of the Middle Kingdom, apparently representing
the setting and rising of constellations, though in an extremely schematic
fashion. The constellations are known as the "decans" because of
their correspondence to intervals of ten days. He furthermore saw the
relationship between these simple pictures and the elaborate representations on
the ceilings of the tombs belonging to kings of the New Kingdom.
It can be safely assumed that the coffin lids are very abbreviated forms
of contemporaneous representations on the ceilings of tombs and mortuary
temples of the rulers of the Middle Kingdom. The logical place for these
representations of the sky on ceilings explains their destruction easily
enough. The earliest preserved ceiling, discovered in the unfinished tomb of
Senmut, the vezir of Queen Hatshepsut, is about three centuries later than the
coffin lids. Then come the wellpreserved ceiling in the subterranean cenotaph
of Seti I and its close parallels in the tomb of Ramses IV and later rulers.
The difficulties we have to face in an attempt to explain these texts can best
be illustrated by a brief discussion of the abovementioned papyrus
"Carlsberg 1." This papyrus was written more than a thousand years
after the Seti text but was clearly intended to be a commentary to these
inscriptions. In the papyrus we find the text from the cenotaph split into
short sections, written in Hieratic, which are followed by a wordforword
translation into Demotic supplemented by comments in Demotic. The original text
is frequently written in a cryptic form, to which the Demotic version gives the
key. We now know, for instance, that various hieroglyphs were replaced by
related forms in order to conceal the real contents from the uninitiated
reader. How successfully this method worked is shown by the fact that one such
sign, which is essential for the understanding of a long list of dates of
risings and settings of the decans, was used at its face value for midnight
instead of evening. It is needless to emphasize what the recognition of such
substitutions means for the correct understanding of astronomical texts. A
complete revision of all previously published material is needed in the light
of this new insight into the Egyptian scheme of describing the rising and
setting of stars the year round. One point, however, must be kept in mind in
every investigation of Egyptian constellations. One must not ascribe to these
documents a degree of precision which they were never intended to possess. I
doubt, for example, very much whether one has a right to assume that the decans
are constellations covering exactly ten degrees of a great circle on the
celestial sphere. I think it is much more plausible that they are
constellations spread over a more or less vaguely determined belt around the
sky, just as we speak about the Milky Way. It is therefore methodically wrong
to use these star lists and the accompanying schematic date lists for accurate
computations, as has frequently been attempted.
The second Demotic astronomical document, papyrus Carlsberg 9, is much
easier to understand and gives us full access to the Egyptian method of
predicting the lunar phases with sufficient accuracy. The whole text is based
on the fact that 25 Egyptian years cover the same time interval as 309
lunations. The 25 years equal 9125 days, which are periodically arranged into
groups of lunar months of 29 and 30 days. The periodic repetition of this
simple scheme corresponds, on the average, very well with the facts; more was
apparently not required, and, we may add, more was not obtainable with the
available simple mathematical means which are described at the beginning of
this section. The purpose of the text was to locate the wandering lunar
festivals within the schematic civil calendar, as is shown by a list of the
"great" and "small" years of the cycle, which contain 13 or
12 lunar festivals, respectively. Accordingly, calendaric problems are seen to
be the activating forces here as well as in the decanal lists of the Middle and
New Kingdom. The two Carlsberg papyri thus give us a very consistent picture of
Egyptian stellar and lunar astronomy and its calendaric relations and are in
best agreement with the level known from the mathematical papyri.
Before leaving the description of Egyptian science, brief mention should
be made of the muchdiscussed question of the "scientific" character
of Egyptian mathematics and astronomy. First of all, the word
"scientific" must be clearly defined. The usual identification of
this question with that of the practical or theoretical purpose of our
documents is obviously unsatisfactory. One cannot call medicine or physics
unscientific even if they serve eminently practical purposes. It is neither
possible nor relevant to discover the moral motives of a scientist—they might
be altruistic or selfish, directed by the desire for systematization or by interest
in competitive success. It is therefore clear that the concept
"scientific" must be described as a question of methods, not of
motives. In the case of mathematics and astronomy, the situation is especially
simple. The criterion for scientific mathematics must be the existence of the
concept of proof; in astronomy, the elimination of all arguments which are not
exclusively based on observations or on mathematical consequences of an initial
hypothesis as to the fundamental character of the movements involved. Egyptian
mathematics nowhere reaches the level of argument which is worthy of the name
of proof, and even the much more highly developed Babylonian mathematics hardly
ever displays a general technique for proving its procedures.
Egyptian astronomy was satisfied with a very rough qualitative
description of the phenomena—here, too, we miss any trace of scientific method.
The first scientific attack of mathematical problems was made in the fifth
century B.C. in Greece. We shall see that scientific astronomy can be found
shortly thereafter in Babylonian texts of the Seleucid period. In other words,
the enormous interest of the study of preHellenistic Oriental sciences lies in
the fact that we are able to follow the development far back into prescientific
periods which saw the slow preparation of material and problems which deeply
influenced the shape of the real scientific methods which emerged to full power
for the first time in the Hellenistic culture. It is a serious mistake to try
to invest Egyptian mathematical or astronomical documents with the false glory
of scientific achievements or to assume a still unknown science, secret or
lost, not found in the extant texts.
III. MESOPOTAMIA
7. Turning to Babylonian astronomy, one's first impression is that of an
enormous contrast to Egyptian astronomy. This contrast not only holds in regard
to the large amount of material available from Mesopotamia but also with
respect to the level finally reached. Texts from the last two or three
centuries B.C. permit the computation of the lunar movement according to
methods which certainly rank among the finest achievements of ancient
science—comparable only to the works of Hipparchus and Ptolemy.
It is one of the most fascinating problems in the history of ancient
astronomy to follow the different phases of this development which profoundly
influenced all further events. Before giving a short sketch of this progress as
we now restore it according to our present knowledge, we must underline the
incompleteness of the present state of research, which is due to the fact that
we do not yet have reliable and complete editions of the text material. The
observation reports addressed to the Assyrian kings were collected by R. C.
Thompson" and in the editions of Assyrian letters published and translated
by Harper, Waterman, and Pfeiffer; much related material is quoted in the
publications of Kugler, Weidner, and others. But Thompson's edition gives the
original texts only in printed type, subject to all the misunderstandings of
this early period of Assyriology, and very little has been done to repair these
original errors. Nothing short of a systematic "corpus" of all the
relevant texts can provide us with the requisite security for systematic
interpretation. The great collection of astrological texts, undertaken by
Virolleaud but never finished, confronts the reader with still greater
difficulties, because Virolleaud composed complete versions from various
fragments and duplicates without indicating the sources from which the
different parts came. And, finally, the tablets dealing with the movement of
the moon and the planets were discussed and explained in masterly fashion by
Kugler; but here, too, a systematic edition of the whole material is necessary.
Years of systematic work will be needed before the foundations for a reliable
history of the development of Babylonian astronomy are laid.
8. Kugler uncovered step by step the ingenious methods by which the
ephemerids of the moon and the planets which we find inscribed on tablets
ranging from 205 B.C. to 30 B.C. were computed. It can justly be said that his
discoveries rank among the most important contributions toward an understanding
of ancient civilization. It is very much to be regretted that historians of
science often quote Kugler but rarely read him; by doing this, they have
disregarded the newly gained insight into the origin of the basic methods in
exact science. This is not the place to describe in detail the Babylonian
"celestial mechanics," as it might properly be called; that will be
one of the tasks of a history of ancient astronomy which remains to be written.
A few words, however, must be said in order to render intelligible the
relationship between Babylonian and Greek methods. The problem faced by ancient
astronomers consisted in predicting the positions of the moon and the planets
for an extended period of time and with an accuracy higher than that obtainable
by isolated individual observations, which were affected by the gross errors of
the instruments used. All these phenomena are of a periodic character, to be
sure, but are subject to very complicated fluctuations. All that we know now
seems to point to the following reconstruction of the history of late
Babylonian astronomy. A systematic observational activity during the Late
Assyrian and Persian periods (roughly, from 700 B.C. onward) led to two
different results. First, the collected observations provided the astronomers
with fairly accurate average values for the main periods of the phenomena in
question; once such averages were obtained, improvements could be furnished by
scattered observational records from preceding centuries. Secondly, from
individual observations, for example, of the moment of full moon or of heliacal
settings, etc., shortrange predictions could be made by methods which we would
call linear extrapolation. Such methods are frequently sufficient to exclude
certain phenomena (such as eclipses) in the near future and, under favorable
conditions, even to predict the date of the next phenomenon in question. After
such methods had been developed to a certain height, apparently one ingenious
man conceived a new idea which rapidly led to a systematic method of longrange
prediction. This idea is familiar to every modern scientist; it consists in
considering a complicated periodic phenomenon as the result of a number of
periodic effects, each of a character which is simpler than the actual
phenomenon. The whole method probably originated in the theory of the moon,
where we find it at its highest perfection. The moments of new moons could
easily be found if the sun and moon would each move with constant velocity. Let
us assume this to be the case and use average values for this ideal movement;
this gives us average positions for the new moons. The actual movement deviates
from this average but oscillates around it periodically. These deviations were
now treated as new periodic phenomena and, for the sake of easier mathematical
treatment, were considered as linearly increasing and decreasing. Additional
deviations are caused by the inclination of the orbits. But here again a
separate treatment, based on the same method, is possible. Thus, starting with
average positions, the corrections required by the periodic deviations are
applied and lead to a very close description of the actual facts. In other
words, we have here, in the nucleus, the idea of "perturbations,"
which is so fundamental to all phases of the development of celestial
mechanics, whence it spread into every branch of exact science.
We do not know when and by whom this idea was first employed. The
consistency and uniformity of its application in the older of the two known
"systems" of lunar texts point clearly to an invention by a single
person. From the dates of the preserved texts, one might assume a date in the
fourth or third century B.C." This basic idea was applied not only to the
theory of the moon (in two slightly modified forms) but also to the theory of
the planets. In this latter theory the main point consists in refraining from
an attempt to describe directly the very irregular movement, substituting
instead the separate treatment of several individual phenomena, such as
opposition, heliacal rising, etc.; each of these phenomena is treated with the
methods familiar from the lunar theory as if it were the periodic movement of
an independent celestial body. After dates and positions of each characteristic
phenomenon are determined, the intermediate positions are found by
interpolation between these fixed points. It must be said, however, that the
planetary theory was not developed to the same degree of refinement as the
lunar theory; the reason might very well be that the lunar theory was of great
practical importance for the question of the Babylonian calendar: whether a
month would have 30 or 29 days. For the planets no similar reason for high
accuracy seems to have existed, and it was apparently sufficient merely to
compute the approximate dates of phenomena, which, in addition, are frequently
very difficult to observe accurately.
We cannot emphasize too strongly that the essential point in the
abovedescribed methods lies not in the comparatively high accuracy of the
results obtained but in their fundamentally new attitude toward the whole
problem. Let us, as a typical example, consider the movement of the sun.
Certain simply observations, most likely of the unequal length of the seasons,
had led to the discovery that the sun does not move with constant velocity in
its orbit. The naïve method of taking this fact into account would be to
compute the position of the sun by assuming a regularly varying velocity. It
turned out, however, that considerable mathematical difficulties were met in
computing the syzygies of the moon according to such an assumption.
Consequently, another velocity distribution was substituted, and it was found
that the following "model" was satisfactory: the sun moves with two different
velocities over two unequal arcs of the ecliptic, where velocities and arcs
were determined in such a fashion that the initial empirical facts were
correctly explained and at the same time the computation of the conjunctions
became sufficiently simple. It is selfevident that the man who devised this
method did not think that the sun moved for about half a year with constant
velocity and then, having reached a certain point in the ecliptic suddenly
started to move with another, much higher velocity for the rest of the year.
His problem was clearly this: to make a very complicated problem accessible to
mathematical treatment with the only condition that the final consequences of
the computations correctly correspond to the actual observations—in our
example, the inequality of the seasons. The Greeks called this a method
"to preserve the phenomena"; it is the method of introducing
mathematically useful steps which in themselves need not be of any physical
significance. For the first time in history, mathematics became the leading
principle for the structure of physical theories.
9. It will be clear from this discussion that the level reached by
Babylonian mathematics was decisive for the development of such methods. The
determination of characteristic constants (e.g., period, amplitude, and phase
in periodic motions) not only requires highly developed methods of computation
but inevitably leads to the problem of solving systems of equations
corresponding to the outside conditions imposed upon the problem by the
observational data. In other words, without a good stock of mathematical tools,
devices of the type which we find everywhere in the Babylonian lunar and
planetary theory could not be designed. Egyptian mathematics would have
rendered hopeless any attempt to solve problems of the type needed constantly
in Babylonian astronomy. It is therefore essential for our topic to give a
brief sketch of Babylonian mathematics.
I think it can be justly said that we have a fairly good knowledge of
the character of mathematical problems and methods in the Old Babylonian period
(ca. 1700 B.C.). Almost a hundred tablets from this period are published: they
contain collections of problems or problems with complete solutions—amounting
to far beyond a thousand problems. We know practically nothing about the
Sumerian mathematics of the previous periods and very little of the interval
between the Old Babylonian period and Seleucid times. We have but few problem
texts from the latter period, but they give us some idea of the type of
mathematics familiar to the astronomers of this age. This material is
sufficient to assure us that all the essential achievements of Old Babylonian
times were still in the possession of the latest representatives of
Mesopotamian science. In other words, Babylonian mathematical astronomy was
built on foundations independently laid more than a millennium before.
If one wishes to characterize Babylonian mathematics by one term, one
could call it "algebra." Even where the foundation is apparently
geometric, the essence is strongly algebraic, as can be seen from the fact that
frequently operations occur which do not admit of a geometric interpretation,
as addition of areas and lengths, or multiplication of areas. The predominant
problem consists in the determination of unknown quantities subject to given
conditions. Thus we find prepared precisely the tools which were later to
become of the greatest importance for astronomy.
Of course, the term "algebra" does not completely cover
Babylonian mathematics. Not only were a certain number of geometrical relations
well known but, more important for our problem, the basic properties of
elementary sequences (e.g., arithmetic and geometric progressions) were
developed. The numerical calculations are carried out everywhere with the
greatest facility and skill.
We possess a great number of texts from all periods which contain lists
of reciprocals, square and cubic roots, multiplication tables, etc., but these
tables rarely go beyond two sexagesimal places (i.e., beyond 3600). A reverse
influence of astronomy on mathematics can be seen in the fact that tables
needed for especially extensive numerical computations come from the Seleucid
period; tables of reciprocals are preserved with seven places (corresponding to
eleven decimal places) for the entry and up to seventeen places (corresponding
to twentynine decimal places) for the result. It is clear that numerical
computations of such dimensions are needed only in astronomical problems.
The superiority of Babylonian numerical methods has left traces still
visible in modern times. The division of the circle into 360 degrees and the
division of the hour into 60 minutes and 3600 seconds reflect the unbroken use
of the sexagesimal system in their computations by medieval and ancient
astronomers. But though the base 60 is the most conspicuous feature of the
Babylonian number system, this was by no means essential for its success. The
great number of divisors of 60 is certainly very useful in practice, but the
real advantage of its use in the mathematical and astronomical texts lies in
the placevalue notation, which is consistently employed in all scientific
computations. This gave the Babylonian number system the same advantage over
all other ancient systems as our modern placevalue notation holds over the
Roman numerals. The importance of this invention can well be compared with that
of the alphabet. Just as the alphabet eliminates the concept of writing as an
art to be acquired only after long years of training, so a placevalue notation
eliminates mere computation as a complex art in itself. A comparison with Egypt
or with the Middle Ages illustrates this very clearly. Operation with
fractions, for example, constituted a problem in itself for medieval computers;
in placevalue notation, no such problem exists, thus eliminating one of the
most serious obstacles for the further development of mathematical technique.
The analogy between alphabet and placevalue notation can be carried
still further. Neither one was the sudden invention made by a single person
but the final outcome of various historical processes. We are able to trace
Mesopotamian numberwriting far back into the earliest stages of civilization,
thanks to the enormous amount of economic documents preserved from all periods.
It can be shown how a notation analogous to the Egyptian or Roman system was
gradually replaced by a notation which developed naturally in the monetary
system and which tended toward a placevalue notation. The value 60 of the base
appears to be the outcome of the arrangement of the monetary units: Outside of
mathematical texts, the placevalue notation was always overlapped by various
other notations, and toward the end of Mesopotamian civilization a modified
system became predominant. It seems very possible, however, that the idea of
placevalue writing was never completely lost and found its way through
astronomical tradition into early Hindu astronomy whence our present number
system originated during the first half of the first millennium A.D.
10. We now turn to the periods preceding the final stage of Babylonian
astronomy which culminated in the mathematical theory of the moon and the
planets described above. It is not possible to give an outline of this earlier
development because most of the preliminary work remains to be done. A few
special problems, however, which must eventually find their place in a more
complete picture, can now be mentioned.
In our discussion of the methods used in the lunar and planetary theories,
we had occasion to mention the extensive use of periodically increasing and
decreasing sequences of numbers. A simple case of this method appears in
earlier times in the problem of describing numerically the changing length of
day and night during the year. The crudest form is the assumption of linear
variation between two extremal values. Two much more refined schemes are
incorporated in the texts of the latest period, but it seems very likely that
they are of earlier origin. Closely related are two other problems: the
variability of the length of the shadow of the "gnomon" and the
measurement of the length of the day by water clocks." The latter problem
has caused considerable trouble in the literature on the subject because the
texts show the ratio 2:1 for the extremal values during the year. A ratio 2:1
between the longest and the shortest day, instead of the ratio 3:2, which is
otherwise used, would correspond to a geographical latitude absolutely
impossible for Babylon. The discrepancy disappears, however, if one recalls the
fact that the amount of water flowing from a cylindrical vessel is not
proportional to the time elapsed but decreases with the sinking level. It is
worth mentioning in this connection that the outflow of water from a water clock
is already discussed in Old Babylonian mathematical texts. This whole group of
texts, however, leads to nothing more than very approximate results. This is
seen from the fact that the year is assumed, for the sake of simplicity, to be
360 days long and divided into 12 months of 30 days each. This schematic
treatment has its parallel in the schemes which we have met in Egyptian
astronomy and which we shall find again in early Greek astronomy; we must once
more emphasize that elements from such schemes cannot be used for modern
calculations, since this would assume quantitative accuracy where only qualitative
results had been intended.
The calendaric interest of these problems is obvious. The same is true
of the oldest preserved astronomical documents from Mesopotamia, the socalled
"astrolabes." These astrolabes are clay tablets inscribed with a
figure of three concentric circles, divided into twelve sections by twelve
radii. In each of the thirtysix fields thus obtained we find the name of a
constellation and simple numbers whose significance is not yet clear. But it
seems evident that the whole text constitutes some kind of schematic celestial
map which represents three regions on the sky, each divided into twelve parts,
and attributing characteristic numbers to each constellation. These numbers
increase and decrease in arithmetic progression and are undoubtedly connected
with the corresponding month of the schematic twelvemonth calendar. It is
clear that we have here some kind of simple astronomical calendar parallel (not
in detail, but in purpose) to the "diagonal calendars" in Egypt. In
both cases these calendars are of great interest to us as a source for
determining the relative positions and the earliest names of various
constellations. But here, too, the strongest simplifications are adopted in
order to obtain symmetric arrangements, and much remains to be done before we
can answer such questions as the origin of the "zodiac."
11. Few statements are more deeply rooted in the public mind or more
often repeated than the assertion that the origin of astronomy is to be found
in astrology. Not only is historical evidence lacking for this statement but
all welldocumented facts are in sharp contradiction to it. All the
abovementioned facts from Egypt and Babylonia (and, as we shall presently see,
also from Greece) show that calendaric problems directed the first steps of
astronomy. Determination of the season, measurement of time, lunar
festivals—these are the problems which shaped astronomical development for many
centuries; and we have seen that even the last phase of Mesopotamian astronomy,
characterized by the mathematical ephemerids, was mainly devoted to problems of
the lunar calendar. It is therefore one of the most difficult problems in the
history of ancient astronomy to uncover the real roots of astrology and to
establish their relation to astronomy. Very little has been done in this
direction, mainly because of the prejudice in favor of accepting without question
the priority of astrology.
Before going into this problem in greater detail, we must clarify our
terminology. The modern reader usually thinks in terms of that concept of
astrology which consists in the prediction of the fate of a person determined
by the constellation of the planets, the sun, and the moon at the moment of his
birth. It is well known, however, that this form of astrology is comparatively
late and was preceded by another form of much more general character
(frequently called "judicial" astrology in contrast to the
"genethlialogical" or "horoscopic" astrology just
described). In judicial astrology, celestial phenomena are used to predict the
imminent future of the country or its government, particularly the king. From
halos of the moon, the approach or invisibility of planets, eclipses, etc.,
conclusions are drawn as to the invasion of an enemy from the east or west, the
condition of the coming harvest, floods and storms, etc.; but we never find
anything like the "horoscope" based on the constellation at the
moment of birth of an individual. In other words, Mesopotamian
"astrology" can be much better compared with weather prediction from
phenomena observed in the skies than with astrology in the modern sense of the
word. Historically, astrology in Mesopotamia is merely one form of predicting
future events; as such, it belongs to the enormous field of omen literature
which is so familiar to every student of Babylonian civilization.
Indeed, it can hardly be doubted that astrology emerged from the general
practice of prognosticating through omens, which was based on the concept that
irregularities in nature of any type (e.g., in the appearance of newborn
animals or in the structure of the liver or other internal parts of a sheep)
are indicative of other disturbances to come. Once the idea of fundamental
parallelism between various phenomena in nature and human life is accepted, its
use and development can be understood as consistent; established relations
between observed irregularities and following events, constantly amplified by
new experiences, thus lead to some sort of empirical science, which seems
strange to us but was by no means illogical and bare of good sense to the minds
of people who had no insight into the physical laws which determined the
observed facts.
Though the preceding remarks certainly describe the general situation
adequately, the historical details are very much in the dark. One of the main
difficulties lies in the character of our sources. We have at our disposal
large parts of collections of astrological omens arranged in great
"series" comprising hundreds of tablets. But the preserved canonical
series come mainly from comparatively late collections (of the Assyrian period)
and were thus undoubtedly subject to countless modifications. We must,
moreover, probably assume that the collection of astrological omina goes back
to the Cassite period (before 1200 B.C.)—a period about which our general
information is pretty flimsy. From the Old Babylonian period only one isolated
text is preserved which contains omina familiar from the later astrology.
Predictions derived from observations of Venus made during the reign of
Ammisaduqa (ca. 1600 B.C.) are preserved only in copies written almost a
thousand years later and clearly subjected to several changes during this long
time. We are thus again left in the dark as to the actual date of the
composition of these documents except for the fact that it seems fairly safe to
say that no astrological ideas appear before the end of the Old Babylonian
period. Needless to say, there are no astrological documents of Sumerian
origin.
The period of the ever increasing importance of astrology (always, of
course, of the abovementioned type of "judicial" astrology) is that
beginning with the Late Assyrian empire. The "reports" mentioned
previously, preserved in the archives of the Assyrian kings, are our witnesses.
But here, again, a completely unsolved problem must be mentioned: we do not
know how the "horoscopic" astrology of the Hellenistic period
originated from the totally different omen type of astrology of the preceding
millennium. It is, indeed, an entirely unexpected turn to make the constellation
of the planets at a single moment responsible for the whole future of an
individual, instead of observing the ever shifting phenomena on the sky and
thus establishing shortterm consequences for the country in general (even if
represented in the person of the king). It seems to me by no means selfevident
that this radical shift of the character of astrology actually originated in
Babylonia. We shall see in the next section that the horoscopic practice
flourished especially in Egypt. It might therefore very well be that the new
tendency originated in Hellenistic times outside Mesopotamia and was
reintroduced there in its modified form. It might be significant that only
seven horoscopes are preserved from Mesopotamia, all of which were written in
the Seleucid period, a ridiculously small number as compared with the enormous
amount of textual material dealing with the older "judicial"
astrology. It must be admitted, however, that the oldest horoscopes known are
of Babylonian origin. On the other hand, at no specific place can all the
elements be found which are characteristic for astrology from Hellenistic times
onward. Neither Babylonian astrology nor Egyptian cosmology furnishes the base
for the fundamental assumption of horoscopic astrology, namely, that the
position of the planets in the zodiac decides the future. And, finally, it must
be emphasized that the problem of determining the date and place of origin of
horoscopic astrology is intimately related to the problem of the date and
origin of mathematical astronomy. Horoscopes could not be cast before the
existence of methods to determine the position of the celestial bodies for a
period of at least a few decades. Even complete lists of observations would not
be satisfactory because the positions of the planets in the zodiac are required
regardless of their visibility at the specific hour. This shows how closely
interwoven are the history of astrology and the history of planetary theories.
V. THE HELLENISTIC PERIOD
12. Before beginning the discussion of the Hellenistic period, we must
briefly describe the preceding development in Greece. Our direct sources of
information about astronomy and mathematics before Alexander are extremely
meagre. The dominating influence of Euclid's Elements succeeded in destroying
almost all references to preEuclidean writings, and essentially the same
effect was produced by Ptolemy's works. Original documents are, of course, not
preserved—one must not forget that even our oldest manuscripts of Greek
mathematical and astronomical literature were written many centuries after the
originals. It is therefore not surprising that our presentday knowledge of
early Greek science is much more incomplete and subject to conjecture than the
history of Mesopotamian or even Egyptian achievements where original documents
are at our disposal. One point, however, can be established beyond any doubt:
early Greek astronomy shows very strong parallelism with the early phases of
Egyptian and Babylonian astronomy, with respect to scope as well as
primitiveness. The astronomical writings of Autolycus" and Euclid"
struggle in a very crude way with the problem of the rising and setting of
stars, making very strong simplifications which were forced upon them by the
lack of adequate methods in spherical geometry. The final goal is again to
establish relations between the celestial phenomena and the seasons of the
years; the problem is thus of essentially calendaric interest. In addition to
these simple treatises, however, we do find one work of outstanding character:
the planetary theory of Eudoxos, Plato's famous contemporary. He made an
attempt to explain the peculiarities of a planetary movement known as
retrogradation by the assumption of the superposition of the rotation of two
concentric spheres around inclined axes and in opposite directions. In this
way he reached a satisfactory explanation of the general type of planetary
movement and thereby inaugurated a new period in the history of astronomy which
was sparked by attempts to explain the movements of the planetary system by
mechanical models. It contains the nucleus for all planetary theories of the
following two thousand years, namely, the assumption that irregularities in the
apparent orbits can be explained as the result of superposed circular
movements. It is only since Galileo and Newton that we know that the circular
orbits do not play an exceptional role and that the great successes of the
Greek theory were merely due to the accidental distribution of masses in our
planetary system. It is, nevertheless, of great historical interest to see how
a plausible initial hypothesis can for many centuries determine the line of
attack on a problem, simultaneously barring all other possibilities. Such
possibilities were actually contained in the approach developed by the
Babylonian astronomers in the idea of superposing linear or quadratic periodic
functions. These arithmetical methods were, however, almost completely
abandoned by the Greek astronomers (at least so far as we know) and survived
only in the treatment of certain smaller problems.
One of these smaller problems is again related to calendaric questions
but also to a basic problem of mathematical geography: the determination of the
geographical latitude by means of the ratio of the longest to the shortest day.
We have already mentioned the Babylonian methods of describing the change in
the length of the days by means of simple sequences. These "linear"
methods reappear in Greek literature and can be followed far into the early Middle
Ages in spite of the invention of much more accurate methods. The term
"linear" does not refer so much to the fact that the sequences in
question form arithmetic progressions of the first order but is intended to
emphasize the contrast with the "trigonometric" method applied to the
same problem and explained in the first book of the Almagest. Here the exact
solution of the problem by the use of spherical trigonometry is given. In
contrast thereto, the linear methods yield only approximate results, but with
an accuracy which was certainly sufficient in practice, especially when one
takes into account the inaccuracy of the ancient instruments used in measuring
time. Historically, however, the main interest lies much less in the perfection
of the results than in the method employed and in its influence on the further
development. A close investigation of early Greek astronomy and
mathematics" reveals an interesting fact. The determination of the time
for the rising and setting of given arcs of the ecliptic, which lies at the
heart of the question of the changing length of day and night, appears to be
the most decisive problem in the development of spherical geometry. It is
typical for the whole situation that a Greek "mathematical" work, the
Sphaerics of Theodosius (ca. 200 B.C.), does not contain a single astronomical
remark. The structure and contents of the main theorems, however, are
determined by the astronomical problem in question; the methods applied
constitute a very interesting link between the Babylonian linear methods and
the final trigonometrical methods.
Trigonometry undoubtedly has a very long history. We find the basic
relations between the chord and diameter of a circle already in use in Old
Babylonian texts which employ the socalled "Thales" and "Pythagorean"
theorems. In sharp contrast to the Greek models for the movement of the
celestial bodies, which operate with circles and therefore necessarily require
trigonometrical functions, we find no applications of trigonometry in the
cuneiform astronomical texts of the Seleucid period which are exclusively based
on arithmetical methods described above.
So far as we know, spherical trigonometry appears for the first time in
the Sphaeric of Menelaos (ca. A.D. 100). The astronomical background of this
work is much more outspoken than in Theodosius, but here, too, much is left to
the reader, who must be familiar with the methods of ancient astronomy to
understand all the astronomical implications. The modern scholar faces an
additional difficulty, namely, the modification of the Greek text by the Arabic
editors. The Greek original is lost, and what we possess is only the Arabic
version made almost a thousand years later. In this interval falls the gradual
transformation of Greek trigonometry, operating with chords, to the modern
treatment, which uses the sine function. It is well known that this change goes
back to Hindu astronomy, where the chords subtended by an angle were replaced
by the length of the halfchord of the halfangle," i.e., our "sin
a." It is, however, a much more involved question to separate these new
methods from those used originally by Menelaos; this question must be answered
if we wish to understand the development of ancient spherical astronomy. This,
in turn, is necessary in order to appreciate the contributions made by the
HinduArabic astronomers which eventually led to the modern form of spherical
trigonometry.
13. It is of great interest to see that the very same problem—the
determination of rising times—leads to still other methods which are now. known
partly as "nomography," partly as "descriptive geometry."
We have a small treatise, written by Ptolemy, called the Analemma. He first
introduces in a very systematic way three different sets of spherical
coordinates, each of which determines the position of a point on the celestial
sphere. Then these coordinates are projected on different planes, and these
planes are turned into the plane of construction, just as we do today in
descriptive geometry. Finally, certain scales are used to find graphically the
relations between different coordinates, again following principles which we
now use in nomography. The Arabs used and developed these methods in connection
with the construction of sundials. Another method of projection, today called
"stereographic," is given in Ptolemy's Planisphaerium. The theory of
perspective drawing in the Renaissance is directly connected with this work.
The practical importance of the determination of the rising times or the
length of the days is not restricted to the theory of sundials. The length of
the longest day increases with the geographical latitude, thus giving us the
means to determine the latitude of a place from the ratio of the shortest to
the longest day. The ratio 3:2 accepted by Babylonian astronomers for the ratio
of the longest to shortest daylight led the Greek geographers to determine
erroneously the latitude of Babylon as 35° (instead of 321°). This error
seriously affected the shape of the eastern part of the ancient map of the
world." The precise relationship can only be established by using
spherical trigonometry, but here, too, the "linear" methods were
applied to various values of the basic ratio in order to give the law for the
changing length of the days for the corresponding latitude. It must be
remarked, however, that at this stage of affairs the concept
"latitude" does not yet actually appear, but the ratio of the longest
to the shortest day itself was used to characterize the location of a place.
Zones of the same ratio were considered as belonging to the same
"clima," a concept which plays a great role in ancient and medieval
geography. The difference in character and behavior of nations living in
different climates furnished one of the main arguments for the influence of
astronomical phenomena on human life.
The second geographical coordinate—the longitude—caused more trouble.
The difference in longitude between two places on the earth is essentially
equivalent to the difference in local time. But there existed no clocks or
signals to compare the local time at fardistant places. Only one phenomenon
could be used as a time signal, namely, records of simultaneous observations of
a lunar eclipse from two different places. If each observer took note of the
local time at which he observed the beginning and end of a lunar eclipse, a
comparison of these records would then furnish the needed information.
Hipparchus proposed the use of this method for an exact construction of the map
of the world, but his program was never carried out. Only one pair of
simultaneous observations seems to have been made, the eclipse of 331 B.C.,
September 20, recorded three hours earlier in Carthage than at Arbela. Actually
the difference in local time between these two localities is much smaller, and
consequently the ancient Map of the world suffers from a serious distortion in
the direction from east to west. Here we see one of the most essential
differences between ancient and modern science at work. Ancient science
suffered most severely from the lack of scientific organization which is so
familiar in our own times. In antiquity, generations passed before a new
scientific idea found a follower able to use and develop methods handed down
from a predecessor. The splendid isolation of the great scholars of antiquity
can only be paralleled with the first beginnings of the new development in the
European Renaissance. It seems to me beyond any doubt that even centers like
Alexandria or Pergamon during their height would appear very poorly equipped if
compared with a modern university of moderate size. And these centers
themselves were few and practically isolated at any particular time; and at all
times they were dependent upon the mood of some autocratic ruler. No wonder
that the great achievements of antiquity are either the result of priestly
castes of sufficiently stable tradition or of a few ingenious men who expended
tremendous energy in restoring and enlarging the structure of a science known
to them from the written legacy of their predecessors. One must not think that
mathematics and astronomy, like the popular philosophical systems or the art of
rhetoric, were taught in the same manner from generation to generation. Three
centuries separate Hipparchus from Ptolemy, one Eudoxos from Euclid, Euclid
from Archimedes and Apollonius. To be sure, the literary tradition was never
interrupted between these outstanding men, but most of the intermediate
literature at best merely preserved and commented. This explains not only why
ingenious ideas were frequently lost (e.g., Archimedes' methods of integration)
but also why it was so easy to destroy ancient science almost completely in a
very short time. Astronomy alone had a slight advantage because of its
practical usefulness in navigation, geography, and timereckoning, supplemented
by the fortunate accident that the Easter festival followed the lunar calendar
of the Near East, thus sanctioning lunar theory when other secular sciences
fell into total desuetude.
The extreme paucity of scientists at almost any given time in antiquity
gave rise to another phenomenon in Greek literature: the publication of
commentaries and popularizing works. A work like the Almagest, written in
purely scientific style, was certainly unintelligible to the majority of people
who needed or wanted to know a modest amount of astronomy. Hence books were
written which attempted to explain Ptolemy's text sentence by sentence, or
which gave abstracts accompanied by explanations of the main principles as far
as this could be done without mathematics. We can observe the same phenomenon
in geography. The first chapter of Ptolemy's Geography" contains a very
interesting theory of map projection, whereas the remaining twelve chapters
constitute an enormous catalogue of localities from all over the then known
world and the corresponding values of longitude and latitude to be plotted into
the network which was to be constructed according to the method explained in
the first chapter. This, again, was not geography for the entertainment of the
general reader. To satisfy popular tastes, there was another literature,
represented by works like Strabo's Geography. These more pleasant writings
furnished serious competition to the strictly scientific literature and
determined to a large extent the character of the field in late antiquity and
the Middle Ages.
14. For the modern historian of ancient astronomy it is therefore of the
greatest value to have an additional source' of astronomical literature in
which the earlier tradition was kept alive without interruption for a much
longer period: the astrological texts. We have already mentioned that astrology
in the modern use of the word appeared very late in antiquity. The art of
casting horoscopes can be said to be a typical Hellenistic product, the result
of the close contact between Greek and oriental cultures. We possess Greek
papyri from Egypt from the beginning of our era to the Arabian conquest showing
us the application of astronomical methods in a great number of specific
horoscopes and in minor astronomical treatises. In addition, an enormous
astrological literature is preserved, catalogued during the last fifty years in
the twelve volumes of the Catalogus by Cumont and his collaborators. Finally,
Vettius Valens, who wrote shortly before Ptolemy, and Ptolemy himself as the
author of the famous Tetrabiblos, must be mentioned.
Modern scholars have not yet made full use of this vast material. The
reason is only too clear: the amount of work to be done surpasses by far the
power of a single individual, and the work itself is certainly not very
pleasant. The astronomical part must be extracted from occasional remarks,
short computations, and similar instances submerged beneath purely astrological
matter of a very unappealing character. But this work must eventually be done
and will give valuable results. As an example might be mentioned the question
of discovering the principle according to which the equinox was placed in the
zodiac. This question must be answered, for on it depend our calculations in
the determination of constellations, chronology, etc. Moreover, systematic
checking of astrological computations will frequently yield information about
the character of the astronomical tables used at the time.
We touch here upon a point of great importance for the modern attitude
toward ancient astronomy. The usual treatment of ancient sciences as a
homogeneous type of literature is very misleading. It is necessary to realize
that very different levels of astronomy or mathematics were coexistent, almost
without mutual contact or interference. One misses the essential points in the
understanding of ancient astronomy if one naively considers various documents
in their chronological order. Even works by the same person must sometimes be
separated from one another. Ptolemy's Almagest is purely mathematical, the
Tetrabiblos (written after the Almagest) is purely astrological, and his
Harmonics" contains a chapter on the harmony of spheres employing concepts
of the planetary movements which contains such strong simplification of the
actual facts that one would try in vain to find similar assumptions in any of
the other works of Ptolemy. In other words, it is necessary to evaluate each
text in its proper surrounding and according to its traditional style. One
cannot, for example, speak without qualification of the contact between
Babylonian and Greek astronomy. Such a contact might even have worked in
opposite directions in different fields. For instance, we have already referred
to the possibility that Hellenistic astrology returned to Babylonia in the form
acquired in Egypt or Syria, whereas observational material from Mesopotamia
undoubtedly influenced Greek mathematical astronomy deeply. In general, it can
be said that the growth of ancient sciences shows much more irregularity and
stratification than modern scientists, accustomed to the fact of the uniform
spread of modern ideas and methods, are prone to assume.
The lack of uniformity in the whole field of ancient astronomy in
general necessarily interferes also with the investigation of any special
problem. We have already mentioned the fact that astrology in the Assyrian age
differed considerably from the horoscopic type which prevailed in late
antiquity and the Middle Ages. But there exists a third type, standing between
the omina type ("when this and this happens in the skies, then such and
such a major event will be the consequence") and the individual birth
horoscope, namely, the "general prognostication," explained in full
detail in the first two books of the Tetrabiblos. This type of astrology is
actually primitive cosmic physics built on a vast generalization of the evident
influence of the position of the sun in the zodiac on the weather on earth. The
influence of the moon is considered as of almost equal importance, and from
this point of departure an intricate system of characterization of the parts of
the zodiac, the nature of the planets, and their mutual relations is developed.
This whole astronomical meteorology is, to be sure, based on utterly naive
analogies and generalizations, but it is certainly no more naive and plays no
more with words than the most admired philosophical systems of antiquity. It
would be of great interest for the understanding of ancient physics and science
in general to know where and when this system was developed. The question
arises whether this is a Greek invention, replacing the Babylonian omen
literature, which must at any rate have lost most of its interest with the end
of independent Mesopotamian rule, whether it precedes the invention of the
horoscopic art for individuals or merely represents an attempt to rationalize
the latter on more general principles. Thus we see that even in a single field
of ancient astronomical thought the most heterogeneous influences are at work;
the analysis of these influences has repercussions on almost every aspect of
the study of ancient civilizations.
15. The same branchingoff into very different lines of thought must
also be recognized in the development of Greek mathematics. The line of
development characterized by the names of Eudoxus, Euclid, Archimedes, and
Apollonius is to be separated sharply from writings like Heron and Diophantus
or the Arithmetic of Nicomachus of Gerasa. Here, again, the question of
oriental influence cannot be discussed as one common phenomenon. Egyptian
calculation technique and mensuration were certainly continued in similar works
in Hellenistic Egypt and found their way into Roman and medieval practices. At
the same time, Babylonian numerical methods influenced Alexandrian astronomy.
How Babylonian algebraic concepts eventually reached Greek writers like Diophantus
is still completely unknown, but that it did is supported by the strong
parallelism in methods and problems. Equally lacking is detailed information as
to the revival of these methods in Moslem literature. On the other hand, the
problems which emerged from the discovery of the irrational numbers are
undoubtedly of Greek origin. It is, however, not correct to consider writings
of the same person as equally representative of "Greek" mathematics.
Those parts of Euclid's Elements (the majority of the work) which deal more or
less directly with the problem of irrational numbers are, as we said before,
Greek. Most likely of equally Greek origin is Euclid's astronomical treatise
called Phenomena, which is written on so elementary a level that nobody would
attribute it to the author of the Elements if the authorship were not so firmly
established. And, finally, Euclid's Data contains the treatment of purely
algebraical problems by geometrical means—which can be interpreted as the
direct geometrical translation of methods well known to Babylonian mathematics.
These methods of "geometrical algebra" in turn determine the whole
structure of Apollonius' theory of conic sections.
Greek mathematics is by far the bestinvestigated field of ancient
science (and of the history of science in general); the situation with respect
to the source material is very good—except where only Arabic manuscripts are
preserved. But one must not forget that also this tradition suffers from severe
gaps. This is due not only to the destruction of manuscripts over a period of
two thousand years but also to the effect of literary influence. I refer not
only to the abovementioned elimination of older treatises by the
overshadowing of the great works of the Hellenistic period. The Greeks
themselves contributed to the distortion of the picture of the actual
development by inventing seemingly plausible stories where the real records
were already lost. The oftrepeated stories about Thales, Pythagoras, and other
heroes are the result. We should now realize that we know next to nothing about
earlier Greek mathematics and astronomy in general and about the contact with
the Near East and its influence in particular. The method which involves the
use of a few obscure citations from late authors for the restoration of the history
of science during the course of centuries seems to me doomed to failure. This
amounts to little more than an attempt to understand the history of modern
science from a few corrupt quotations from Kant, Goethe, Shakespeare, and
Dante.
16. Undoubtedly the most spectacular advances in the history of
astronomy untilvery recent times were scored in the theory of the planets. The
catchwords "Ptolemaic" and "Copernican" refer to different
assumptions as to the mechanism of the planetary movement. This is not the
place to underline the fact that the Copernican theory is by no means so
different from or so superior to the Ptolemaic theory as is customarily
asserted in anniversary celebrations, but we must briefly analyze Ptolemy's own
claims to having been the first one who was able to give a consistent planetary
theory. This claim seems to contradict not only the existence of prePtolemaic
planetary tables in Roman Egypt as well as in Mesopotamia but also Ptolemy's
own reference to such texts. What Ptolemy means, however, becomes clear if one
reads the details of the introduction to his own theory. He requires an
explanation of the planetary movement by Means of a combination of uniform
circular movements which refrains from simplifications like the assumption of
an invariable amount for the retrograde arc and similar deviations from the
actual observations. Indeed, in order to remain in close agreement with the
observations, Ptolemy had to overcome difficulties which Hipparchus was not
able to master and which led Ptolemy to a model which is very close to Kepler's
final solution of the problem, by assuming not only an eccentric position of
the earth but also an eccentric point around which the movement of the
planetary eccenter appears to be uniform. The resulting orbit is of almost
elliptical shape with these two points as foci. This whole theory is closely
related in method to the explanation of the "evection" of the moon (a
periodic perturbation of the moon's orbit discovered by Ptolemy) by a combination
of eccentric and epicyclic movements. Both theories are real masterpieces of
ancient mathematical astronomy which far surpassed all previous results.
It is not surprising that Ptolemy's results overshadowed all previous
works. All that we know about his forerunners comes mainly from the Almagest
itself. We hear that Hipparchus used eccenters and epicycles for the
explanation of the anomalies in the movement of the sun and the moon,'"
and we learn about theorems for such movements proved by Apollonius.'"
This brings us to the very period (about 200 B.C.) from which the oldest
cuneiform planetary texts are preserved—computed, however, on entirely
different principles. These cuneiform texts cover the two centuries down to the
time of Caesar. A direct continuation, chronologically speaking, but of still
another type, are planetary tables from Egypt, written in Demotic or Greek.
These tables give the dates at which the planets enter or leave the signs of
the zodiac. Such tables were known to Cicero"' and are most likely the
"eternal tables" quoted with contempt by Ptolemy. We do not know how
these tables were computed, and their occurrence in Greek as well as in Demotic
leaves us in doubt as to their origin—showing us only the degree of
interrelation we can expect in Hellenistic times.
The most interesting question would, of course, be to learn more about
Hipparchus' astronomy. He is most famous as the discoverer of the precession of
the equinoxes. Though this fact cannot be doubted,'" underlining its
importance lays the wrong emphasis on a phenomenon which gained its importance
only from Newton's theory, which showed that precession depends on the shape of
the earth and thus opened the way to test the theory of general gravitation by
direct measurements on the earth. For ancient astronomy, however, precession
played a very small role, requiring nothing more than sufficiently remote and
sufficiently reliable records of observations of positions of fixed stars. The
change in positions must then eventually become evident; and little difficulty
was encountered in incorporating this slow movement into the adopted model of
celestial mechanics. What we actually need to appreciate in Hipparchus'
contribution must be derived from a careful study of all relevant sections of
the Almagest, not by the schematic method of obtaining "fragments"
from direct quotations but by a comparison of Ptolemy's methods and the older
procedures which he frequently mentions. That such an approach can lead to
welldefined results has recently been shown in the theory of eclipses.
17. One of the most important problems in connection with Hipparchus is,
of course, the problem of the dependence of Hipparchus (and Greek astronomy in
general) on Babylonian results and methods. Whatever the conclusions derived
from a deeper knowledge of Hipparchus' astronomy may turn out to be, one thing
is clear: the century between Alexander's conquest of the Near East and
Hipparchus' time is the critical period for the origin of Babylonian
mathematical astronomy as well as for its contact with Greek astronomy. Since
Kugler's discoveries, which showed the exact coincidence between numerical
relations in cuneiform tablets and in Hipparchus' theory, no one has doubted
Babylonian priority. It is an undeniable fact that the Babylonian theory is
based on mathematical methods known already in Old Babylonian times and does
not show any trace of methods considered to be characteristically Greek. The
problem remains, however, to answer the question: What caused the sudden
outburst of scientific astronomy in Mesopotamia after many centuries of a
tradition of another sort? On what background can we understand, for example, the
report that the "Chaldaean" Seleucus from Seleucia on the Tigris
completed the heliocentric theory, previously proposed as a hypothesis by
Hipparchus? Greek influence on late Babylonian astronomy must not be denied or
asserted on aprioristic grounds, if we really want to understand a phenomenon
of great historical significance.
These remarks are not intended to make Greek influence alone responsible
for the new developments in Mesopotamia. As a matter of fact, this answer would
only raise the equally unsolved question why Greek astronomy suddenly emerged
from many centuries of primitiveness to a scientific system. The alternative,
Greek or Babylonian, might even exclude the right answer from the very
beginning. It also seems possible that the rise of mathematical astronomy in
Hellenistic times resulted from the suddenly intensified contact between
several types of civilization, in some respects to be paralleled with the
origin of modern science in the Renaissance. In other words, neither the Greeks
nor the Orientals might have been alone responsible for the new development but
rather the enormous widening of the horizon of all members of the culture of
the Hellenistic age. One result of this process was probably the new attitude
toward the relationship between the individual and the cosmos, expressed in the
new form of horoscopic astrology. In this case it is quite evident that Egypt
and Greece—and perhaps Syria as well—contributed about equally much to the
refinement and spread of this new creed. It is equally possible that the
contact between Greek scholars, trained to think in geometrical terms which
Greek mathematics had developed in the fifth century, and Babylonian
astronomers, equipped with superior numerical methods and observational
records, brought into simultaneous existence two closely related types of
mathematical astronomy: the treatment by arithmetical means in Babylonia and
the model based on circular movements in the Greek centers of learning in the
eastern Mediterranean. It may well be that competition, not borrowing, was the
chief contributor to the initial impetus. At any rate, it is clear that each
detail in the development of Hellenistic astronomy which we will be able to
understand better will reveal a new aspect in the fascinating process of the
creation of the new world which was destined to become the foundation of the
Roman and medieval civilizations.
The unique role of the Hellenistie period in the field of sciences, as
in other fields, can be described as the destruction of a cultural tradition
which dominated the Near East and the Mediterranean countries for many
centuries, but also the founding of a new tradition which held following
generations in its spell. The history of astronomy in the Hellenistic age is
especially well suited to demonstrate that the great energies liberated by the
disintegration of an old cultural tradition are very soon transformed into
stabilizing forces of a new tradition, which includes about as many elements of
development as of stagnation.
VI. SPECIAL PROBLEMS
18. Every research program in a complex field will face the need of
constant modification and adjustment to unforeseen complications and new
ramifications. Problems can arise and results be obtained without having been
anticipated in the original question. The context of a mathematical text, for
example, can determine with absolute certainty the meaning of a word otherwise
only vaguely defined; signforms in a papyrus which is exactly dated by
astronomical means may furnish valuable information for purely paleographical
problems. From dates and positions given in Demotic astronomical texts, it
follows that the Alexandrian calendar introduced by Augustus was used by
Egyptian scribes only a few years after the reform, very much in contrast to the
common opinion that the Egyptians were especially conservative in general and
in calendaric matters in particular. In short, from few, but solidly
established, facts we can learn more than from all general speculations.
One of the problems which at first sight lies very much outside the
history of ancient astronomy is the study of social and economic conditions of
the ancient civilizations. There are, however, several points of contact
between these studies and astronomy. We are indebted to Cumont for a masterly
investigation of the information contained in the astrological literature from
Hellenistic Egypt. His results are not only of interest for the history of
ancient civilization but also illustrate very well the background of the men
who used and transmitted the astronomical material known to us from the
planetary tables or from Vettius Valens. It turns out that the soil in which
these practices were rooted was essentially Egyptian, in spite of the use of
the Greek language in the documents. This is in perfect harmony with the close
parallelism between Greek and Demotic planetary texts mentioned above and shows
the constant interaction of Greek and native influences in Hellenistic Egypt.
It also shows how dangerous it is to decide the authorship of Hellenistic
doctrines or methods simply on the basis of such superficial grounds as the
language used.
The analogous question for Babylonia seems to be easier to answer. The
Mesopotamian origin of the astrological omina cannot be doubted. We would,
however, like to know more about the background of the astronomers of the
latest period. It is well known that the names of three Babylonian astronomers
appear in Greek literature and that two of them actually were found on
astronomical tablets, though in an unclear context. For one particular place,
the famous city of Uruk in South Babylonia, we can go much further. It can be
shown that the scribes and owners of our texts belong to one of two
"families," or perhaps "guilds," of scribes who frequently
call themselves scribes of the omenseries "Enuma Anu Enid."' We can
follow the work of these scribes very closely for almost a hundred years until
the school of Uruk ceased to exist, probably because of the Parthian invasion
of Babylonia in 141 B.C. In contrast thereto, the school of Babylon survived
the collapse of the Greek regime, as is proved by a continuous series of
astronomical texts down to 30 B.C. This is an interesting result in comparison
with the assumption that Babylon practically ceased to exist after the Parthian
occupation. The grouping of our texts according to welldefined schools is also
of interest from another point of view. It can be shown that two different
systems of computation existed side by side for a long time. Competing schools
of this sort constitute a phenomenon which is usually considered characteristic
for Greek culture.
19. Countless thousands of business documents are preserved from all
periods of Mesopotamian history. For the urgently needed investigation of
ancient economics, a precise knowledge of the metrological systems is of the
greatest importance. Unfortunately, the scientific study of Babylonian measures
has been sadly neglected. Fantastic ideas about the level and importance of
astronomy in the earliest periods of Babylonian history led to theories which
brought measures of time and space in close relationship with alleged
astronomical discoveries. We know today that all these assumptions of the early
days of Assyriology must be abandoned and that Babylonian metrology must be
studied from economic and related texts clearly separated according to period
and region. For the determination of Old Babylonian relations between various
measures, the mathematical texts are of great value because they contain
numerous examples which give detailed solutions of problems in which
metrological relations play a major role. The consequences of such relations,
established with absolute certainty, are manifold. For example, we now know
from Old Babylonian mathematical texts the measurements of several types of
bricks as well as the peculiar notation used in counting bricks. It is evident
that such information is of importance for the understanding of contemporary
economic texts dealing with the delivery of bricks for buildings, thus leading
to purely archeological questions. Metrological relations are also needed if we
wish to gain an insight into wages and prices. Returning to our subject, it
must be said that metrology is of great importance not only for the history of
the economics of Mesopotamia but also for purely astronomical problems.
Distances on the celestial sphere are measured in astronomical texts by units
borrowed from terrestrial metrology. The comparison between ancient observation
and modern computations thus requires a knowledge of the ancient relations
between the various units. This problem is by no means simple because our
astronomical material belongs to relatively late periods, Assyrian and
NeoBabylonian, and the metrological system of these times is much more involved
than the Old Babylonian. Mathematical texts would certainly be of great help
here too, but the few tablets from this period are so badly preserved that they
present us with at least as many new questions as they answer. NeoBabylonian
economic texts will therefore furnish the main point of departure for the study
of the latest phase of Mesopotamian metrology and its astronomical
applications.
It might be mentioned, in this connection, that theories about direct
relationship between early Mesopotamian metrology and astronomy also gave rise
to the rather unfortunate concept of high accuracy in the determination of
weights, measures of length, etc. It is of great importance to realize that the
absolute values of all metrological units are subject to great margins of
inaccuracy and local and temporal variations. The first step in a historical
investigation of Mesopotamian metrology must therefore be to establish from
economic and mathematical texts the ratios between the units; these ratios have
an incomparably better chance of showing unformity than the absolute values
deduced from accidental archeological finds.
20. Closely related to metrological problems is the question of the
accurate identification of ancient star configurations. Much work remains to be
done before it will be possible to give a reliable history of the topography of
the celestial sphere in general, or even of the zodiacal constellations. In
spite of attempts to make Egypt responsible for many forms, the predominant
influence of Babylonian concepts on the grouping of stars into pictures must be
maintained. But neither Babylonian nor Egyptian developments are known in
detail. The identification of Egyptian constellations is especially difficult,
mainly because it must be based on relations between the times of rising and
setting and therefore depends on elements which are grossly schematized in the
texts at our disposal. The situation in Mesopotamia is slightly better because
we have actual observations in addition to the schematic lists, at least for
the later periods which are of special importance for the Hellenistic forms of
the constellations.
For the period following the publication of the Almagest, we must take
into account the possibility of still other complications. We know from
explicit remarks in the Almagest that Ptolemy's star catalogue introduced
deviations from older catalogues. Astrological works, however, may very well
have maintained prePtolemy standards both with respect to the boundaries of
constellation and the counting of angles in the zodiac. We have already
mentioned the stubborn adherence of astrological writers to methods of
computation which were made obsolete by the development of spherical
trigonometry. For the modern historian it is therefore of importance to
establish the specific standard according to which a given document was
written, especially when chronological problems are involved.
21. While metrology is a muchneeded implement for economic history and
the understanding of ancient astronomy, astronomy itself serves general history
in chronological problems. Chronology is the necessary skeleton of history and
owes its most important fixed points to astronomical facts. We need not
emphasize the use of reports of eclipses, especially solar eclipses, for the
determination of accurate dates to form the framework into which the results of
relative chronology must be fitted. It must be underlined, however, that the
available material is by no means exhausted. A better understanding and
reinvestigation of the reports of the Assyrian astronomers will certainly
furnish new information of chronological value. It must be stated, on the other
hand, that not too much is to be expected from older material. In order to make
ancient observations accessible to modern computation, a certain degree of
accuracy must be granted; this accuracy seems to be missing in the earlier
phases of the development of astronomy. This, for instance, makes the older
Egyptian material so ill suited for chronological purposes. For later periods,
however, Egypt has furnished and will furnish much information from
astrological documents. It is particularly calendaric questions, such as the
use of eras and similar problems, which have been illuminated by the dating of
horoscopes.
The great variety of calendaric systems, local eras, and older methods
of dating raises many difficulties in ancient chronology. This difficulty was
clearly felt also by ancient astronomers and was the cause of the early use of
consistent eras in Babylonian and Greek astronomy. The Babylonian texts always
use the Seleucid Era, whereas Ptolemy reduces all dates to the Nabonassar Era
but uses the Old Egyptian years of constant length. This crossing of Egyptian
and Babylonian influences is paralleled by the subdivision of the day into
hours. The Egyptians divided the day into twelve parts from sunrise to sunset,
thus obtaining hours whose length depended on the season. The Babylonian
astronomers used six subdivisions of day and night, but these units were of
constant length. Combining the Egyptian division into 24 hours with the
Babylonian constancy of length, the Hellenistic astronomers used
"equinoctial" hours for their computations and solved the problem of
finding the relationship between seasonal and equinoctial hours by spherical
trigonometry. One sees here again what a multitude of relations, problems, and
methods contributed to shape concepts such as a continuous era or the 24hour
day which are so familiar to us today.
Ancient chronology and the accurate analysis of ancient reports have
turned out to be of interest even to a modern astronomical problem. In 1693
Halley discovered the fact that the moon's position appeared to be advanced
compared with the expected position as computed from positions recorded by
Ptolemy. This "acceleration" can be explained by a slow increase in
the length of the solar day or by a decrease in the rotational velocity of the
earth. Such a decrease is caused by tidal forces, and it is of great interest
to determine the amount as accurately as possible. For this purpose, accurate
positions of the moon in remote times are of great value, and such positions
can, indeed, be derived from records in cuneiform texts. Modern measurements of
high precision can thus be supplemented by observations in antiquity.
22. Not only are Hellenistic astronomy and Hellenistic astrology the
determining factors for the astronomy and astrology of the Middle Ages in
Europe, but its influence is equally important for the development of
astronomical methods and concepts in the Middle and Far East. We must therefore
at least mention an enormous field which still awaits systematic research:
Hindu science. This does not mean that there is not an extensive literature on
this subject; indeed, even a small number of original texts are published. The
main trouble lies, however, in the tendency of the majority of publications by
Hindu authors to claim priority for Hindu discoveries and to deny foreign
influence, as well as in the opposite tendency of some European scholars. This
tendency has been especially strong so far as Hindu mathematics is concerned,
and it is aggravated by the inadequate publication of the original documents,
from which usually only scattered fragments are cited in order to prove some
specific statement. As a result, there is no means today to obtain an
independent judgment from the study of the original texts which are preserved
in enormous number, though of relatively late date for the most part.
The situation with respect to Hindu astronomy is not much better. There
can be little doubt that the original impetus came from Hellenistic astronomy;
the use of the eccentricepicyclic model alone would be sufficient proof even
if we did not also find direct witness in the use of Greek terminology. This
fact is interesting in itself, but it may very well be that the period of
reception lies between Hipparchus and Ptolemy; systematic study might therefore
reveal information about prePtolemaic Greek astronomy no longer preserved in
available Greek sources. Hindu astronomy would in this case constitute one of
the most important missing links between late Babylonian astronomy and the
fully developed stage of Greek astronomy represented by the Almagest.
The fundamental difficulty in the study of Hindu astronomy lies in the
character of the preserved textual material. The published and commented texts
consist exclusively of cryptically formulated verses giving the rules for
computing certain phenomena, making it extremely difficult to understand the
actual process to be followed. It is evident, on the other hand, that no
astronomy of an advanced level can exist without actually computed ephemerids.
It must therefore be the first task of the historian of Hindu astronomy to look
for texts which contain actual computations. Such texts are, indeed, preserved
in great number, though actually written in very late periods. Poleman's
catalogue of Sanskrit manuscripts in American collections lists about a hundred
such manuscripts in the D. E. Smith collection in Columbia University in New
York. In their general arrangement, these texts are reminiscent of the
cuneiform ephemerids from Seleucid times and must reveal many details of the
Hindu theory of the planetary movement if attacked by the same methods which
have proved so successful in the case of the Babylonian material. The complete
publication of this material is an urgent desideratum in the exploration of oriental
astronomy.
As mentioned above, the texts in the D. E. Smith collection are of very
recent origin, only a few centuries old. This does not mean that the methods
used are not of very much earlier date. This is shown by the investigation of
one of these texts, which deals with the problem of the varying length of the
days during the year. Though written about 1500, the computations are based on
methods going back to a much older period. Analogous results can be expected in
the remaining material, and there is no reason to assume that the D. K Smith
collection exhausts all the preserved material.
23. In the preceding sections we have frequently touched on
methodological questions. In closing, I wish to underline a few principles in a
more general way. As is only natural, the study of the development of ancient
science began under the influence of the ancient tradition. Herodotus,
Diodorus, the commentators of Plato, etc., were the sources which determined
the picture of the early stages of Greek and oriental mathematics and
astronomy. But while students of political history, art, economics, and law
learned in the early days of systematic archeological research to consider this
literary tradition about the ancient Orient as nothing more than a supplementary
source to be checked by the original documents, the majority of historians of
the exact sciences have remained in a stage of naïve innocence, repeating
without criticism the nursery stories of ancient popular writers. This is all
the more surprising because many of these stories should have revealed their
purely fictitious character from the very beginning. Every invention considered
of basic importance is attributed to a definite person or nation: Thales
"discovered" that a diameter divides the area of a circle into two
equal parts, Anaximandes and several others are credited with the discovery of
the obliquity of the ecliptic, the Egyptians discovered geometry, the
Phoenicians arithmetic—and so on, according to an obvious pattern of naïve
restoration of facts the origins of which had been totally forgotten. Modern
authors then add stories of their own, such as the idea that the construction
of the pyramids required mathematics, the assumption of supposedly marvelous
skies of Mesopotamia, and the notion of Egyptian Stone Age astronomers
industriously determining the heliacal rising of Sirius or carrying out a geodetic
survey of the Nile Valley.
It is clear that the replacement of the traditional stories by
statements based exclusively on results obtainable from the original sources
will not be very appealing. This is the inevitable result in the development of
every science; for increased knowledge means giving up simple pictures. In the
history of science, an additional element must be added to the steady increase
of complexity resulting from a better understanding of our sources. Not only do
we learn to interpret our material more accurately but we also learn to see
everywhere the immense gaps in our preserved sources. We will more and more be
forced to admit that many, and essential, steps in the development of science
are hopelessly destroyed; that we, at best, are able to sketch mere outlines of
the history of science during certain sharply limited periods; and that many of
the driving forces might actually have been quite different from those which we
customarily restore on the analogy of later periods.
One consequence of this situation seems to me to be evident: unless the
history of science now enters the stage of specialization, it will lose all
value in the framework of historical research. It must be clearly understood
that the history of science must work with methods and must consider its
problems from viewpoints which correspond to the methods and standards of other
branches of historical research. The idea must definitely be abandoned that the
history of science must adapt its level to the alleged requirements of the
teaching of the modern fields of science. The intrinsic value of this research
must be seen in its contribution to our understanding of the historical
processes which shaped human civilization, and it must be made clear that such
an understanding cannot be reached without the closest contact with the other
historical fields. The call for specialization is not very popular. I am
convinced, however, that a wellfounded insight into the details of a single
essential step in the development is at present of higher value and more
fascinating than any attempt at general synthesis. It is ridiculous to believe
that we are anywhere able to reach "final" results in the study of
the development of human civilization. But the overwhelming richness of all
phases of human history can be appreciated only if we occupy ourselves with the
real facts as accurately as possible and do not attempt to hide their manifold
aspects under the veil of hazy generalizations or let our judgment be guided by
the naive idea of human "progress." Every synthesis written fifty
years ago is now completely antiquated and at best enjoyable for its literary
style; the careful study of the original works of the ancients, however, will
reveal to everyone and at any time the development of their achievements.
The call for specialization must not be misunderstood as a plea for the
disregard of the general outlines of the historical conditions. On the
contrary, specialized work can be accomplished successfully only if the points
of attack are selected under constant consideration of possible interference
from other problems and other fields. It is indeed the most gratifying result
of detailed research on a welldefined problem that it necessarily uncovers
relationships which are of primary importance for the understanding of larger
historical processes. The actual working program, however, needs restriction
and minute detail work. The most essential task is that of making the original
sources accessible as easily as possible in their best available form. By the
indefatigable work of Heiberg, Hultsch, Tannery, and many others, we possess
today a great part of the extant writings of the Greek scientists in excellent
editions. We owe to Sir Thomas Little Heath many brilliant commentaries and
translations of Greek mathematicians. To make Greek and oriental source
material more generally accessible, supplemented, of course, by modern
translations and commentaries, will be the foremost problem of the future. The
extension of this program to include medieval material, on the one hand, and
Middle Eastern documents, on the other, appears as a logical consequence,
worthy of the serious efforts of all scholars who wish to contribute to the
understanding of the past of our own culture.
BROWN UNIVERSITY
