John Pazmino
 NYSkies Astronomy Inc
 2010 November 14
    On 12-13 November 2010 the 'Mathematician's journey' conference met 
in New York at the Institute for the Study of the Ancient World. It 
was cosponsored by many mathematics and archaeOlogical societies, 
including Courant Institute, Courant carried the meeting in its own 
75th anniversary celebrations and furnished three speakers for it. 
    The conference commemorated the work of Otto Neugebauer, a 
mathematician and historian who built the present discipline of 
ancient mathematics in the 1920s and 1930s. While there are numerous 
articles and single lectures about Neugebauer since he passed away in 
1990, the 'Mathematician's journey' meeting was the first real 
convention dedicated to him. The convention is commonly shortnamed as 
the 'Neugebauer conference'. 
    Notice of the meeting went to scholars and groups dedicated to 
mathematics and astronomy, as well as to archaeologists and 
historians. NYSkies received an invite for home astronomy in the New 
York City region. 
    NYSkies posted the announcement in the NYSkies Yahoogroup and the 
November 2010 issue of NYC Events. The latter circulates thruout the 
astronomy world in the NYSkies website and Internet astronomy 
    The convention was free of charge and included the opening of a 
parallel exhibit 'Before Pythagoras'. It, also at the Institute, runs 
thru 17 December 2010. On November 11th an opening reception for the 
exhibit was held for the conventionists, which I attended. 
    In this article I explain some circumstances about Babylonian 
astronomy as taken from the convention speakers and discussion with 
attendees. In no way can I elaborate on the astronomical mathematics 
of the Babylonians in this article! There are many good references for 
this subject, such as those by Otto Neugebauer himself 
    The Institute for the Study of the Ancient World, ISAW (IGH-saw) 
is a unit of New York Univeristy to bring together separate endeavors 
to understand the workings of early civilizations. It was set up in 
2007 and is housed in a converted mansion at 15 East 84th Street, 
between Fifth and Madison Avenues. It is down the block from the 
Metropolitan Museum of Art, with which ISAW has deep collaboration, 
and is a couple blocks from downtown Yorkville. 
    The house is tastefully refurbished from is original use as a 
residence. It is common all over Manhattan for such older structures 
to be recycled into homes for cultural and civic societies. The house 
at 9 East 84th St, closer to Fifth Avenue, is currently in renovation 
for some other such group. 
    The renovation is not faithful to the earlier motif, but fitting 
none the less. The public areas has mod cons, like new lighting, fire 
qunenchers, elevator, and so on. The basement, where the only open 
restrooms were, was glatt contemporary with the usual machinery and 
furnishings of any small office edifice. 
    The exhibit was in a first floor room where the public can visit 
with minimal disturbance to the offices around and above it. The 
conference and reception were on the second floor, reached by curved 
stairs or an lift. The receptions and breaks were in the library 
facing 84th Street with views of the Metropolitan from its windows. 
The conference was in an auditorium facing the middle spine of the 
84th-85th Street block. This shielded it from street noise and made it 
easier to darken for projecting the slides. 
    The only clumsy facility were the restrooms. There were none on 
the second or the ground floor. You had to go two floors down, by 
elevator, to the cellar. There were no obvious stairs reaching below 
the ground floor. The restrooms were unisex, with both gender symbols 
on the doors. 
    NYSkies had the pleasure to hear ISAW's Dr Alexander Jones discuss 
'Babylonian and Greek astronomy' on 16 September 2009. He fielded many 
good and serious questions from the audience and continued the dialog 
at the postlecture dinner. 
Otto Neugebauer
    There being a deep litterature about Neugebauer, here I give a 
brief outline of his life to guide you for further inquiry. I here 
must describe a stuh-RANJ feature of the conference. 
    The meeting consisted of scholars, academics, scientists, many of 
whom knew or worked with Neugebauer. Yet, I heard his name pronounced 
in many ways. It being German, the sound is like 'NOY-geh-ba-werr' 
with a hard 'g' for 'golf'. I heard among the speakers sounds of NOO-
geh-ba-werr, NAU-geh-ba-werr, NOW-geh-ba-werr, NIGH-geh-ba-werr, NAY-
geh-ba-werr! Sometimes during a one talk the pronunciation shifted! 
    Neugebauer was born in Austria in 1899 and attended college there 
until 1921. He studied engineering, being that his father worked for 
the Austria railroad agency. 
    He in 1921 entered University of Munich to continue engineering, 
but soon developed an interest in mathematics. In 1922 he moved to 
University of Gottingen to study under its director Richard Courant. 
Neugebauer and Courant became close social colleagues at Gottingen. 
    Richard Courant years later came to America and founded the 
Courant Institute, a cosponsor of the meeting. In spite of his maths 
profession Neugebauer seems to have only one article on mathematics, 
differential equation of quasiperiodic functions, to his name. 
    At Gottingen his interest drifted to history of mathematics to the 
extent of founding a publication in 1929 'Quellen und studien' to 
provide copies of source texts for others to study. He worked on this 
with assyriologists and archaeologists. 
    He first studied Egyptian texts and then those from Babylonia. 
These were unearthed during the many Germany-based digs in the 
Babylonia region. From these Neugebauer discovered a depth of maths 
not before appreciated and built up a discipline to investigate it. .   
    By 1933 troubles began in Germany under new leader Adolf Hitler. 
Neugebauer and thousands of other German scientists and academics were 
drummed out of their careers. He left For Denmark in 1933.
    In Copenhagen he shifted gears again to ancient astronomy and 
found that existing work was scattered in many unrelated places. In 
1936 he began to consolidate older texts and many newer ones. 
    His efforts were curtailed by worsening conditions in Germany, 
affecting his colleagues and publishers. He resigned his ties with 
Germany in 1938. Courant already left Germany for a mathematics career 
at new York University, later to establish the Courant institute. 
    In 1939 Neugebauer moved to the US for a professor of mathematics 
at Brown University, He never returned to Germany again. He tossed 
German for his writing, using there after only English. 
    In 1951 he wrote 'The exact sciences in antiquity' for the lay 
audience, summarizing the mathematical basis of astronomy in Babylonia 
and Egypt. Older home astronomers grew up with this book, as a Dover 
paperback, and was permeated with lifelong interest in the subject. I 
sure did, making my attendance at the conference just an other 
instance of that interest. 
    At Brown he built up its history of sciences section and founded a 
department of egyptology. He continued writing and publishing about 
ancient astronomy, now his main theme. Among his works were 
'Mathematical cuneiform texts' in 1945 and 'Astronomical cuneiform 
texts' in 1955. Both incorporated large numbers of pieces from 
American musea not easily obtainable in prewar Europe. Later works 
include 'Egyptian astronomical texts' in 1969 and 'History of ancient 
mathematical astronomy' in 1975. 
    In his later years, Neugebauer branched out to ancient cultures 
elsewhere, like Ethiopia, early Christianity, Mediaeval Europe. 
    In the 1970s he spent semesters at Princeton's Institute for 
Advanced Study. In 1984 he moved to there, where he stayed until his 
death in 1990. 
    At Princeton he discovered that Babylonian methods in astronomy 
were still in wide use in the West into the current era. One example 
is the use of Babylonian day-length tables far from the latitude of 
Babylonia, where they are badly wrong. This is now a major corridor of 
study, the transmission of ancient science to modern times. 
    NYSkies supports and encourages scholarly work like the Neugebauer 
conference. That it was held in New York is a credit to the City's 
role as a world center for astronomy, both campus and home. By having 
a home astronomy representation, with a couple NYSkies attendees, the 
unity of the profession, with no phony dichotomy between  'amateurs' 
and 'professionals', is strengthened. 
    NYSkies contributed take-aways for the meeting, left in the 
library for the several breaks between rounds of talks. One was a map 
and description of Babylonian stars in the autumn sky of 2000 BC. 
    Attendees were at first confused because the sky outside, in 
autumn of 2010, looks nothing like the map. They then realized the 
effect of precession! While Pegasus, around the vernal equinox, is 
near the meridian at nightfall today, the vernal equinox was in Taurus 
back then. With the vernal equinox on the meridian back then, the 
stars of Taurus and other 'winter' constellations were around it. 
    The other was a copy of my article on Babylonian full and new 
Moons, from the NYSkies website. Dr Jones kept his own copy from his 
talk before NYSkies and noted that he uses it for inquiries about 
Babylonian lunar observations. Attendees liked this piece but some 
were weak on the astronomy. I helped them as they asked. 
    After a couple talks I noticed a really odd feature. All of the 
slideshows, all being digital, were from the same mold! They were 
plain text or simple pictures with no bells or whistles. 
    The text was large, clear, and crisp, of a legible typestyle, 
either dark on white field or negative. Pictures occupied most of the 
slide, sometimes with simple wording added. There were no audio or 
video segments and very modest use of color (except for pictures 
already in color). Slide were flipped, like a chemo slide projector. 
    It was as if the speakers gave their source material to a one 
person, who assembled the shows for them with a single template. An 
alternative is that a template was issued to the speakers, who then 
dropped their source items into it. This is radicly different from the 
usual conference when each speaker has his own show with widely 
distinct motifs and style. 
    While this means of preparing the shows was possible, I will be 
surprised if it was the way they were made. The conference was no 
different than any other academic meeting where the speakers, all from 
colleges or institutes, have local help to compose their slideshow. 
    In the final session on Saturday afternoon speaker Dr Dennis Duke 
gave a slideshow of more original form. He had more varied typestyles, 
colorful backgrounds, freer layout for pictures. 
    The room was small enough for a calm audience to hear the speaker 
acousticly. A microphone was provided at the lectern. There was some 
room noise from ventilation but not overly disturbing. Street noise 
was abated by distance. 
    Babylonia is a complex of peoples, divvied into several 
territories. The sections had distinct language, culture, lifestyle. 
They sometimes warred on each other. For this piece, I combine them 
into one culture centered on Babylon, near present day Baghdad, Iraq. 
    This culture is also called Mesopotamia, from the Greek 'between 
rivers'. The rivers were the Tigris and Euphrates that flow northwest 
to southeast thru modern Irag. The headwaters are in Kurdistan, Syria, 
and Turkey, causing no end of dispute about allocation and withdrawals 
of water among these countries. 
    The land between and near the rivers is flat, with remote bounding 
mountains, allowing for ample farm and irrigation. The Babylonians are 
among the first hydrology engineers in human history. This flat land 
was made of a fine silt from the upper waters that settled into vast 
plains of clay. This clay was the material for, among other items, 
bricks, pottery, artwork, and writing tablets. 
    The time frame is the first and second millennium BC, with ups and 
downs in fortunes. The Babylonian culture faded with the conquest by 
Persia and then by Greece and Rome. In spite of the sometimes scratchy 
life, much of the core of mathematics and astronomy endured into the 
Roman era and much is in routine use even today. 
    Virtually all of naked-eye astronomy as taught to newcomers into 
the profession is taken recta mente from the textbooks used in those 
far ago centuries. Our concepts of planet aspects, lunar motions, 
eclipses, calendar, day-night cycle, angle and time measures, are 
bonds between us astronomers of the 21st century and those of the 10th 
and earlier centuries before the common era. A current almanac of 
celestial events does look pretty much like one unearthed from Uruk. 
Cuneiform writing 
    Mesopotamians wrote on slabs, plates, pads made from the river 
clay, there being no dependible supply of suitable fiber, like 
papyrus. The clay was flattened into sheets of a size needed to hold 
its inscriptions or into palm-sized shells or glands. The latter were 
flat on the obverse, front, side for writing and domed or convex on 
the reverse, back, to fit the hand for a secure grip. 
    We find these clay pieces of assorted sizes from huge murals to 
small pads. The 'Before Pythagoras' artifacts are the latter.Their 
almost dainty quality surprised some attendees used to seeing only the 
large specimina. 
    To write on soft clay, before it hardens in air or is baked. the 
writer pressed a stylus into the clay. He could not really 'draw' like 
on paper because the clay yielded under the stylus. 
    The stylus was a reed with a bevel or chisel point at one end and 
a point at the other. With a clay pad nestled in one hand the stylus 
was worked with the other hand to make wedge-shaped marks, These look 
somewhat like arrowheads. The length and orientation of these marks 
combined to form words and phrases. 
    The writing from the wedge patterns is cuneiform, from Latin 
'cuneus', wedge. The term was used for a wedge of cake or pie and for 
seats in a round stadium. The sound is 'koo-NAY-ih-form', not 'KOO-
    Clay is a soft pliable material. Mistakes can be smoothed out and 
bits of filler clay can be added. After the inscription is finished 
the piece is fired to dry and harden it, like for pottery or bricks. 
    In many cases it appears that the clay pieces were baked in a 
conflagration. These occurred by sincere accident or thru warfare. I 
suspect the pieces in 'Before Pythagoras' came from an incendiary 
incident. The pads are from school exercises, unlikely to be preserved 
on purpose. 
    The dried pieces are brittle and can be broken with the slightest 
provocation. This is evidenced by the dominance of fragments, more 
than complete tablets, in museum collections. Some pads in the exhibit 
had broken edges and effaced text. 
Cataloging texts
    Cuneiform texts are almost all in musea, with a good sample in 
private hands. They are cataloged within the museum's inventory system 
with an alphanumeric string of characters, like 'YBC 10529' and '55-
24-357' for two pieces in the exhibit. 
    A piece may have several designations according as the collection 
or inventory system was modified by the museum. Since older 
litterature referenced the piece by its former designation, the 
scholar must understand them as well as the current name. 
    There is no uniform method of naming the texts. Each museum has 
its own scheme, which may be that for other segments of its 
collection. Many texts have common or proper names within the jargon 
of the scholars. When discussing a piece, the catalog number and the 
host museum should be specified. 
    When a piece is published in transcription or translitteration, 
it is often referenced by the item in the publication. The 
publication is far more available to scholars than the original piece. 
Deciphering texts 
    Deciphering the cuneiform material started in the early 1800s and 
was well developed by the late 1800s. Most of the work was done in 
Germany, a situation prevailing thru the 1930s and resumed in the 
1950s. If you want to study Mesopotamia you better learn German to 
read the articles and understand vocabulary. 
    The first astronomy related texts were recognized in the 1880s as 
lists of stars, location of planets, calendar tables. There was little 
'maths' in these, just numbers like the angular distance of Jupiter 
from Antares on a given date or the date when Venus first appeared out 
of morning twilight. 
    Be the 1910s inscriptions turned up with genuine maths work in 
them. These were practical problems like figuring the number of bricks 
needed for a wall of a given dimensions or the volume of a certain 
shape of tank. These showed a mastery of geometry with good values of 
pi and other irrational numbers. 
    More astronomy texts came in the 1920s onward with instructions 
for computing assorted events. These were mostly simple in their maths 
yet cunning in the theory and practice. One amazing find was the method 
to compute the illuminated fraction of the Moon for a given date. 
    The deciphering is normally in three steps. The initial step is is 
transcription, a straight copying of the cuneiform marks from the 
original into a clearer form far easier to inspect. This scheme is 
also cheaper and simpler to publish than photographs of the original 
tablets. The intent is to provide an authentic rendering of the source 
in a form suitable for dissemination for other scholars. 
    This transcription is done by hand. It is not possible, nor 
prudent, to do a rubbing, like on stone inscriptions. It takes skill 
to correctly read the marks. They are sometimes distorted before the 
clay hardens or are broken off at the fracture lines after hardening. 
    Another problem is sloppy writing by the original scribe, like bad 
caliigraphy today. And the scribe did make honest copying errors. 
    Only trusted scholars may work with the slabs personally during a 
visit to the home museum. Others may work with photographs. A proposed 
transcription is reviewed by a second scholar before it is released. 
Even so, once in a while articles point our mistakes in interpreting a 
text, resulting an in new meaning. 
    It could be possible to construct replicas by the laser profiling 
technique. This is used by musea for, as example, fossils and bones. 
The replicas are exact copies of the outer contour and texture of the 
original, allowing many studies, measurements, comparisons to be done 
as if the scholar had in hand the originals. As at 2010 this is not 
done for the clay inscriptions. 
    Other factors are the lack of word separations and of vowels,  
both common features in ancient writing, It would, for example, be 
tricky to read English if all the words were run together with no 
spaces, vowels, punctuation. 
    When working with a damaged original, missing or illegible parts 
are drawn as hash or stipple. An author may guess at a partial word 
and enter it with a footnote. 
    The transcription doesn't have to be full size. It can be a size 
convenient for publication. A scale of length is included in the 
drawing or caption. 
    An incredible realizations for the newcomer to ancient astronomy 
is the tiny fraction of inscriptions read by modern scholars. Musea 
are filled with perhaps a million samples but only a few thousand were 
ever put into transcription. There simply are too few scholars trained 
for this excruciatingly tedious task. 
    The next step is translitteration, a conversion of the cuneiform 
marks into something that can be pronounced in a modern language. 
Since this is usually German, the sounds can be strange to the 
American astronomer. Yet it is a uniform scheme across assyriology. 
   The translitteration is not only plain Latin letters. It has 
numbers and orthographic marks mixed in. A pronunciation scheme was 
devised as a convention, not to reproduce the original, and generally 
unknown, sounds from Babylonia. 
    The third step is translation into clear modern text. In as much 
as the original may miss out words or phrases, the clear text may be 
interrupted with words added in to fill the gaps. You can not expect 
smooth and clean prose. You'll read a series of bursts of words. like 
notes taken for a lecture or class. 
    In fact, many texts are in this style, probably from the tedium of 
writing in clay. The words on the piece may be just bits and pieces, 
words or phrases, more like reminders for the contemporary reader. 
Number system 
    It took many decades to figure out the number system in the 
cuneiform texts. By the 1920s it was realized the numbers are a mix of 
decimal and sexagesimal notation. In the place of distinct symbols for 
each of the 59 digits in the base 60, the digits were written in 
decimal form. A one is a vertical stroke with a wedge at the top like 
a 'Y'. A ten is a deep left-pointing wedge like the left beak '<'. 
    From these tow symbols all numbers are built. To conserve space, 
the symbols were stacked in groups of three, represented here by a '/' 
to indicate the level. Examples are 32 = <<< YY, 8 = YYY/YYY/YY, 59 = 
<<</<< YYY/YYY/YYY 
    For numbers above 59, a place-value system was used, just like our 
decimal system. The same pair of symbols < and Y, were notched one 
column to the right or left to show bands of 60s or 1/60s. 
    Cuneiform has no 'decimal point' or zero. This makes it really 
tough to figure out the value of a number except by context. The 
number YYY could mean 3, 180 (3 * 60), or 1/20 (3 * 1/60) among 
    Cracking the number system was facilitated by the diaries and 
almanacs for astronomy. In them numbers commonly run in a sequence. 
After some scrutiny, the pattern was uncovered and sued to unravel the 
meaning of the symbols. 
    In translations commas separate the groups of 60 and a semicolon 
separates the integer from fraction like a 'decimal point'. If YYY  
really means 3/60, we write  0;3. 3 is 3 and 180 is 3,0. Some modern 
authors convert the numbers into decimal notation for easier reading. 
This destroys intelligence hidden in the original notation. 
    This base-60 number scheme was handed down civilization by 
civilization to us today. We use it for our time and angle measures as 
degrees or hours, then 1/60 minutes, and 1/3600 seconds. In earlier 
years the division was carried into the thirds, fourths, fifths, and 
higher, like carrying a decimal to more places. 
    Babylonian is the first place to work with real graphs. They were 
simple X-Y plots from which values were read out. As example we have a 
graph of the depth of immersion into the Earth's shadow by the Moon 
versus the angular distance from a lunar node. By this the extent of a 
lunar eclipse was taken out given where the eclipse took place along 
the Moon's path. 
    The function in the graph was never higher than a linear or ramp 
function. The compiler figured out the maximum and minimum value of 
the Y-parameter and the cycle of its variation of the X-value. Then 
connected the these points by straight lines. As crude as this was it 
was an stunning advance in human thought. 
    In addition to the prime quantity, they built graphs of the first 
and sometimes second differences of this parameter. In this way they 
accounted for varying 'speed' of the Y-value versus the X-value. This 
amounts to an incipient form of derivatives in calculus. 
    The graphs were poorly drawn due to the yielding quality of clay 
under a stylus trying to make straight lines. In some samples we see 
the lines were made in short strokes, point by point. 
    I can not in this conference summary give a deep treatment of the 
astronomy practiced in Babylonia. I offer some extra items to help 
fill out your other reading. 
    Probably the most compelling fact to come from studying Babylonian 
astronomy is that it shows the first time in human history that purely 
objective means were applied to describe and predict natural events. 
There was no edifice of supernatural being, spirits, deities to power 
the stars and planets. 
    While the Mesopotamians had their deities, they were outside of 
science. Yet the strategy of applying maths to nature didn't catch on. 
Following cultures, thru Greece and Rome and Mediaeval times, called 
on spirits and deities to explain natural happenings.. 
    It wasn't until Galileo's time that maths was successfully applied 
to any other field of nature, notably kinetics and mechanics. This put 
humanity of the road to Einstein physics. 
    One obstacle for earlier Meospotamian studies in astronomy was the 
lack among historians for easy competent calculation or simulation of 
ancient skies. Dialog with astronomers was scratchy and often the 
historian obtained his needed computations by favors. 
    Historians made do with what ever their campus library had to 
hand, occasional visits to the campus observatory, calculation by 
approximate methods. The results were not too confident and were 
presented sometimes as suggestion. 
    Universities acquired mainframe computers in the 1950s, at first 
for management and clerical tasks. Gradually faculty was fitted with 
terminals to communicate to the central computer. Departments in 
science and mathematics grabbed onto computers right away. The 
mathematical capabilities of the machines was not fully exploited by 
the 'soft' departments, like history and antiquities study. 
    In the 1990s home computers grew powerful and cheap enough to be 
standard equipage in scholar's offices. They were tied to the campus 
network facilities, by which Internet access was provided. Historians 
could spontaneously query colleagues for astronomy assistance. 
    Thee computers also were ample enough for high accuracy 
simulations of remote past time thru planetarium software. NYSkies 
heard a presentation in July 2010 about a solar eclipse in Homer's 
Odyssey. The speaker, Marcelo Magnasco of Rockefeller University, 
examined it by planetarium programs like those on a typical home 
astronomer's computer. 
    With thee new tools, work with ancient astronomy advanced rapidly. 
For example, we learned that many alleged celestial events, like 
marking the birth of a certain ruler, never took place. They were made 
up to validate the ruler's office. In other cases we demonstrated that 
an event treated as a fable in fact took place. It then helped fix the 
date of the associated story. 
    We plain do not know about the individuals or groups who practiced 
astronomy in Babylonia. The notion of assigning discovery, invention, 
development to specific persons started with the Greeks. We speak of 
Euclides geometry, Plato geography, Hipparchus astronomy, Apollonius 
spheres, and so on. We have nothing like this for astronomy or other 
science and maths for Babylonians. 
    In Mesopotamia the best we can do is note where an inscribed slab 
was found by town or court. The texts don't tell who wrote them, 
supervised the work, contributed data for them. 
    This is a feature of other facets of Babylonian society. Only the 
highest level of society are named, the kings and military leaders.    
Perhaps because of this lack of attribution, so much a part of Greek 
and our present culture, Mesopotamian culture was in earlier scholarly 
studies sometimes downplayed or undervalued. 
    There was, and still is to a significant degree, disagreement 
about the observational skills of ancient peoples. Given that they had 
only the naked-eye and simple instruments, could they secure data 
accurate enough to develop a respectable system of planetary motions? 
    It is surprisingly easy to get longitude and latitude of a planet 
relative to a zodiac star to within one degree consistently. This is 
good enough for building a theory for the planets and Moon. 
    The Babylonians needed a coordinate system based on the ecliptic, 
which we still use today as the ecliptic system of lat-lon. They 
measured from the instant vernal equinox, wholly beyond cognizance of 
precession, in the eastward direction. To simplify their calculations 
the zodiac was sectioned in to 12 signs, each of 30 degrees. 
    They signs were named for the zodiac constellations they lined up 
with, more or less, and the vernal equinox was plotted at the front, 
west, of sign Taurus. Over the centuries of the 2nd and 1st millennium 
BC the vernal equinox slided westward thru into Aries by the end of 
the Babylonian era. We have records of it being, as example, in the 
13th degree of Aries, in the middle, not beginning, of a sign. 
    It migrated to the first degree of Aries in about 150BC, where the 
Greeks stabilized it until the present day. We let the signs migrate 
with the vernal equinox. They no longer match the constellations. 
    A very confusing aspect of Mesopotamian astronomy was the use of 
the lunar calendar,normalized into 12 mean lunar months. Each month, 
for easier maths, was cut into 30 parts, each a tithi (TIH-thee). This 
is almost one degree or one day of solar motion. It is shorter enough 
to throw off modern analysis based on the true degree. 
    There were several kinds of almanac published by the Babylonians. 
One is a diary of planets and Moon observations. These are usually 
measurements of a planet relative to a star or Moon and may have notes 
about missed observations due to weather. They also describe eclipses, 
comets, unusual meteors, and possibly aurorae. 
    An other is a reference table of computed events, based on the 
model for the particular event. These are not observations as such and 
the real moment of the event can be off by a day or two. These were 
used as lookup tables for calculations. 
    The third is the prediction for a future event. These were based 
on the model for the event. The second and third almanacs are 
distinguished by their internal or contextual date. 
    Of special value in historical investigation of Babylonian 
astronomy are the 'procedure' texts. These are instructions or 
examples of various computations, sometimes associated with the 
reference tables. The instructions sometimes include facts and figures 
not obvious in the tables to show additional Mesopotamian skill. 
    The Babylonians recognized certain stars in the zodiac against 
which they assayed the location of the planets, Sun, and Moon. I don't 
try here to reproduce the Babylonian names for these stars, but refer 
to them and constellations by translations and modern equivalents. 
    Star names and groupings evolved over the 2,000 year span of 
Babylonian society. What I give here is only one example set of names. 
Expect other sources to differ from this. 
    The special zodiac stars are the  Normal Stars, picked about as 
best as practical given the irregular disposition of suitable stars in 
the sky. They are the same as those we would casually use to track 
planet motions and. Many are still today posted in almanacs for planet 
and Moon conjunctions. 
    Contemporary tables of these stars started with the vernal equinox 
and continued downrange thru the zodiac signs. In the earliest period 
the equinox was near Aldebaran. This star was easily recognized by the 
Hyades cluster and position about midway between Orion and Pleiades. 
It was also reddish in tint. 
    By sheer luck the opposite point in the ecliptic was marked by 
Antares, quite six signs away. It, too, was in an easily recognized 
field of stars and has a reddish hue. The two stars today still 
balance on opposites sides of the ecliptic. 
    Babylonian 'Normal Stars' 
    bet Ari, alp Ari
    eta Tau, alp Tau, bet Tau, zet Tau 
    mu Gem, gam Gem, eta Gem, alp Gem, bet Gem
    the Cnc, del Cnc, gam Cnc
    eps Leo, alp Leo, rho Leo, the Leo 
    bet Vir, gam Vir, alp Vir
    alp Lib, bet Lib
    bet Sco, del Sco, alp Sco
    the Oph 
    mu Sgr, xi Sgr (uncertain) 
    bet Cap, gam Cap, del Cap 
    --- Aqr (none known) 
    eta Psc 
    There is a nasty gap of some 50 degrees from Capricornus thru 
Pisces with no known Normal Stars. I suppose stars in the Circlet of 
Pisces and Water Jug of Aquarius can fill the gap, but apparently not. 
    More surprising, there seems to be none in Sagittarius, with 
several bright stars near the ecliptic. The two listed above, in small 
asterisms, are not fully confirmed, but do show up occasionally in the 
Babylonian diaries. 
    The use of gamma and delta Cancri as benchmarks in the zodiac 
implies that the Mesopotamians knew the Beehive cluster. We have no 
certain references to it as yet. On the other hand, they recognized 
the Coma cluster as the Frond, a large fancy leaf. 
    The Mesopotamian astronomers knew the Andromeda galaxy! It was 
treated as a separate star, the Rainbow. I'm not making this up! This 
makes the Babylonians the first humans to show cognizant awareness of 
a really deep sky object, an other galaxy. 
    The star coordinates were always ecliptic lat-lon, there being no 
evidence that an equatorial system was known. In the earlier times the 
longitude scale was fixed in the stars. Later it was tied to the 
vernal equinox to participate in precession. 
    We don't know where the Mesopotamian zodiac came from. They seem 
to originate within Mesopotamia, not handed down from an other 
culture. The Greeks modified the names to fit their own folklore and 
mythology, where they stayed to this day. 
    The correspondence between the Babylonian signs and the Greek and 
Roman ones is shown here. We today use the Roman names, in Latin. 
        Babylonian     | Greek         | Roman       | English 
        Hired Man      | Krios         | Aries       | Ram 
        Bull of Heaven | Tauros        | Taurus      | Bull 
        Great Twins    | Didymoi       | Gemini      | Twins  
        Crab           | Carkinos      | Cancer      | Crab 
        Lion           | Leon          | Leo         | Lion 
        Furrow         | Parthenos     | Virgo       | Maiden 
        Balance        | Kheli         | Libra       | Balance 
        Scorpion       | Skorpios      | Scorpius    | Scorpion 
        Forefather     | Toxotes       | Sagittarius | Archer 
        Goat-fish      | Aigokeros     | Capricornus | Goat-fish 
        Great One      | Hydrokhoos    | Aquarius    | Water-bearer 
        Tails          | Ikhthyes      | Pisces      | Fishes 
    The entire heavens accessible from Mesopotamian latitudes was 
partitioned into groups of stars as early-day constellations. They 
were named for features in the culture. Mythology, on the Greek level 
wasn't fully matured, but many groups were named for deities that the 
people appealed to. Stars within these groups were often named for 
their anatomical location. 
    The number of groups is loose because over the ages some groups 
were combined, others altered, others abandoned. The total number was 
about 50, about as many as the classical constellations seen from a 
similar latitude. The sky had its voids, pretty much where the Greeks 
had them, like in Camelopardalis and Monoceros and Sculptor. 
    Note well that the Greeks did not just take over the Babylonian 
star groups and apply their own names and stories to them. Many are in 
fact transplants into Greek while others are rearrangements. It is not 
really correct to say that such-&-such Babylonian group became the 
Greek other-such constellation. The two cultures independently 
partitioned the sky. 
    The translitterated names are strange to modern astronomers and 
are not fully stable from scholar to scholar. Modern descriptions tend 
to use only the translated names. 
    Since the Babylonian society endured for about two thousand years, 
we can expect that their star names and patterns evolved. There is no 
one and only one Babylonian uranography. Examples in our lifetime in 
the mid 20th to early 21st century illustrate this trait. 
    * alp Cen's name Toliman is still only lightly cited 
    * alp CrB is both Alphecca and Gemma about equally 
    * alp Per was Algenib, then became Mirfak 
    * bet Cen is both Agena and Hadar about equally 
    * gam Cas was Cih,then became Navi 
    * gam Dra is both Etamin and Eltanin  
    * eta UMa was Benetnash, then became Alkaid 
    * Camelopardalis is sometimes called Camelopardus 
    These are only a dew specimina of how nomenclature can evolve and 
do so quickly. Similar mutations took place in early societies with 
Babylonia as no exception. 
    This is the name for a peculiarly Babylonian calendar table. These 
were drawn up in tabular or wheel figure. In the latter form, the 
tablet is erroneously called an astrolabe, but there was never any 
mechanical operations built into it. 
    The concept of the three-stars-each is that in each month three 
stars or asterisms were selected to mark the month by their heliacal 
rising. There were 'three stars each' for the month. 
    Heliacal rising was an important event in the course of a star or 
planet. The rising marks the first time the body is seen in dawn 
twilight after rounding conjunction with the Sun. On the day before 
the heliacal rising the body rises in a twilight too bright to reveal 
it by the proximity of the Sun beyond it. 
    On the next day the Sun moved on an extra degree, enough to allow 
the target to shine thru a twilight not quite so bright. As the sky 
continues to rotate twilight brightened to immediately snuff out the 
body. On days after then the target is in sight for a longer and 
longer period with no special further note taken of it. 
    In the modern scheme of planet motions the heliacal rising marks 
the beginning of an apparition. Nowayears we don't compute or observe 
the heliacal rising. In the stead we assign a nominal interval after 
conjunction. This ranges from a week to a month, depending on the 
target's brightness. Alternatively we note as a recognition the day 
that the target rises at civil or nautical dawn. 
    We don't know why three stars were used, but it may be redundancy 
in case of adverse sky conditions. The idea was that when the stars 
were observed in their heliacal risings the new month was in progress. 
This was in addition to marking the beginning of the month by the 
appearance of the Moon in dusk after new Moon. 
    The three stars for a month were lined up left to right along the 
horizon as a 'north', 'middle', and 'south' star. We don't know for 
sure if the stars were in three declination bands or if the terms are 
relative, in place of a left-middle-right notation. 
    By reconstructing the sky for the era of a given table or wheel, 
we figured out what the stars and groups were. In some examples other 
stars and groups are recorded.  In a few instances a planet was 
included among them. Maybe that inscription was for a particular year. 
    There also seem to be some inconsistencies, like including the Big 
Dipper. This asterism was semperpatent in Babylonia with no distinct 
helical rising. 
    Study of the three-stars-each calendars from different centuries 
reveals the effects of precession, on top of probable revised choices 
of stars or different groupings of the stars. In this we find nothing 
at all of any awareness of the precession mechanism. 
    Babylonia knew all of the naked-eye planets and followed their 
wanderings in the zodiac. Babylonia never had a kinematic or spatial 
construction of the cosmos like the Greeks did. 
    In all of their work, the Mesopotamians neglected latitude. Only 
the planet longitudes mattered. This is an approximation commonly 
employed today among astronomers to get rough-&-ready answers to 
assorted problems of planetary motion and position. 
    They recorded the dates of alignments of the planets relative to 
stars and the Sun. They recorded also the synodic aspects we still use 
today: stations, conjunctions, oppositions. Modern almanacs still 
dutifully note these very same aspects. 
    In additional to the planet deployments the Babylonians monitored 
the heliacal risings of the planets in the dawn twilight. We sometimes 
deal with this situation as first visibility of a planet, announcing 
its incoming apparitions or seasonal appearance. 
    From their observational records they worked out the sidereal and 
synodic periods to a credibly good accuracy and could predict the 
occurrence of future alignments to within a day or two. 
    For the occasions when the place of a planet was needed between 
synodic events, a linear interpolation was applied. This method took 
into account the varying speed of the planet thru the zodiac as a 
correction to the place found by mean motions. 
    In all of this work the Mesopotamians shown no inkling of 'orbit' 
'center of motion', 'perihelion/perigee' or any other construct of the 
heavens. The planets were points sliding on the celestial sphere. 
    Only for the Moon did they realize any sense of distance. the Moon 
was closer to the Earth than the stars and planets, proved by their 
occultation by the Moon. For solar eclipses the Moon covered the Sun. 
At lunar eclipses she passed thru the shadow of Earth, while none of 
the planets or stars did. 
    There appears to be no awareness of the varying brightness of a 
planet. Mars and Mercury vary in brightness widely, from good and 
bright to dim and obscure. It was only the planet's motion and 
position that mattered. 
Sidereal and synodic cycles
    One of the really cunning abilities of the Babylonians was to 
recognize the interplay between a planet's sidereal and synodic 
cycles. The sidereal cycle is the interval between successive returns 
of the planet to the same place in its orbit against the stars. But 
the Mesopotamians had absolutely no concept of this definition and 
could not in any feasible way observe the sidereal cycle. 
    The synodic cycle is the interval between successive returns of 
the planet to the same elongation from the Sun or to the same synodic 
event. This was recta mente observed and recorded. 
    This latter definition is sometimes missed in modern astronomy. 
Synodic intervals hold true for ANY position relative to the Sun, not 
just conjunction or opposition. The occurrence of an elongation of 45 
degrees west from the Sun are one synodic cycle apart. 
    The key to unlocking the mystery of the planet motions is the 
equality of a certain number of sidereal to synodic periods, the both 
occupying an integer number of years. Since a year is the sidereal 
period of Earth, the relation is 
    (Earth sid cycles) = (planet sid cycles) + (planet syn cycles) 
The numbers are the count of cycles, not the duration of the cycle. 
The former is a pure numeric; the latter, a time unit.  
    This formula holds true for ANY given span of Earth years, not 
just integer intervals. The count of the planet cycles is also 
noninteger. Since it is easy to count off synodic intervals by direct 
observation, the sidereal interval is a derived quantity. 
    The above relation is equivalent to the standard one employed in 
modern astronomy 
    (planet syn period) = ((plan sid per) * (Earth sid per)) 
                         / ((plan sid per) - (Earth sid per)) 
Because Earth's sidereal cycle takes is one year, this reduces to 
    (planet syn period) = (plan sid per) / ((plan sid per) - (1)) 
with the answer in Earth years. Notice that an inner planet has a 
negative synodic period, the usual convention. 
    It is easy to assert that the long spans of time needed to get all 
three cycles to be integers means that the Babylonians continuously 
observed for millennia far into the past. That they needed observed 
positions and motions of the planets is certain, but not for hideously 
long spans of time. 
    It takes only a few centuries of observation, plus a leap of faith 
that the cycles are stable for the indefinite future, to suss out the 
relation. Also, it was good enough to get the relation to hold for a 
few more centuries, given the precision of measurement of quite one 
degree in longitude. 
    For one sample of how the cycles of a planet were derived, I look 
at Jupiter. By observation, Jupiter runs thru 10.9884 synodic cycles 
in 12 years. Since the sidereal period can not be monitored directly, 
it came from the above equation 
    (Jup sid cyc) = (Earth years) - (Jup syn cyc) 
                  = (12) - (10.9884) 
                  = (1.0116) 
    After 12 full years Jupiter completes a bit more than one full 
sidereal cycle and runs over into the next by 4.176 degree. 
    Also by observation, after 71 Earth years, Jupiter completes 
65.0145 synodic cycles. From the formula this equals 5.9855 sidereal 
cycles. He fall short of the 6th full cycle by 5.2200 degree. 
    We build a table of trials: 
    Years | syn cyc | sid cyc  | sid arc | excess 
       12 |  10.988 |  1.0116 |   4.176 | +4.1760 
       71 |  65.015 |  5.9855 | 354.780 | -5.2200 
       83 |  76.002 |  6.9971 | 358.956 | -1.0440 
       95 |  87.000 |  8.0088 |   3.168 | +3.1680 
      166 | 152.006 | 13.9943 | 357.948 | -2.0820 
      261 | 228.997 | 22.0030 |   1.080 | +1.0800 
      344 | 315.000 | 29.0002 |   0.072 | +0.0720 
      427 | 391.003 | 35.9973 | 359.028 | -0.9720 
    It looks like a round of 12 years plus a following one of 71 years 
will about cancel out the excess. This is the entry for 83 years (12 + 
71). It also seems that four rounds of the 83-year cycle plus one 12-
year cycle can about erase the error. This is the entry for 344 years 
(4 * 83 + 12). I entered a few more cases that almost balance out the 
over and under fall. Some were noted in Babylonian inscriptions. 
    Once a set of integral sidereal and synodic cycles is established 
a table of synodic phaenomena can be drawn up to cover every instance 
during this interval, say 344 years. The table then cycles back to the 
initial alignment and begins a second series. 
    The Babylonians had only to establish the relation between 
sidereal and synodic cycles thru observations of only a few centuries. 
They then constructed by pure number work a reasonable span for the 
Jupiter events to repeat. 
    This master table has the dates and ecliptic longitude of the 
synodic events for each year in the cycle. For the 344-year cycle, 
there are 344 years, each with unique dates and degrees. They 
gradually slip year to year until by the 345th year they align again 
with the first year. 
    If an astronomer asked when Jupiter next has an eastern station 
at Spica he first looked at current observations of Jupiter. Say an 
opposition occurred at Aldebaran in 2010. (This is made up.) He looked 
up the Aldebaran event in the master table and counted up the years 
from there until the next Spica event. Say this was 103 years. He 
added 103 to the initial year of 2010 to get 2113 for the next eastern 
station next to Spica. 
    In substance this is exacta mente how Babylonians sussed out solar 
and lunar eclipses. The length of the series is 'saros', just their 
word for 'duration' or 'period'. After one saros the circumstances for 
an eclipse recur, with stipulated odds of being seen from Mesopotamia. 
    Like a wooden nickel the question of awareness of precession by 
ancient astronomers comes up. Hipparchus is the first to clearly 
describe the effect and apply it to older observations of the stars. 
Were there earlier recognitions of precession? 
    It seems so far there is not. Yet, every culture lasting more than 
a few centuries and carrying out astronomy observations was affected 
by precession. In their monuments and inscriptions they casually note 
changes in the location of the cardinal ecliptic points and alignment 
of buildings. In many cases, structures were replaced or rebuilt to 
shift alignment after it wandered too far off thru precession. 
    Yet, no one upped and declared what's really going on. The best we 
find is a vague sense that 'something funny' happened to the stars. In 
the case of Mesopotamia, we have examples of instructions that note 
the place of the equinox or solstice. In later centuries the location 
moves farther west along the ecliptic, with no realization of just 
what is causing the shift. 
    Overall, within the era of written history, precession at a given 
latitude shoves stars flanking the autmunal equinox colure southward, 
to lower south altitude or beyond the south horizon. Stars bordering 
the vernal equinox colure are pulled northward to expose new stars 
over the south horizon. 
    As luck has it, the pushed off stars are bright ones in Centaurus-
Crux that shine thru the thick air on the horizon and were mapped in 
earlier times. The stars lifted into view are mostly dim ones in 
Sculptor-Phoenix. These wee often missed in early mapping. 
    An intriguing discovery in recent years is that buried in the data 
accumulated by the Babylonians was a distinct sidereal and tropical 
year. According as the purpose and means of determination there were 
many values for the length of the year. All were close, within a few 
minutes, and some seem to be rounded. 
    The length was taken as either star-to-star or equinox-to-equinox 
with no appreciation that the two must be different due to precession. 
The sidereal year is about 20 minutes longer than the tropical year, 
an amount easily hidden in the moise of measurements. 
    The currency of many lengths with dispersion of several minutes 
thoroly smeared out the distinction of the two lengths and smothered 
all chance to discover the precession effect. 
    It is only now thru modern study of the texts that we uncovered 
the two different years. They are consistent with a precession drift 
of 1.36 degree per century, virtually spot on with the modern value. 
Lunar motions
    The acme of Babylonian astronomy was in the theory of the Moon. In 
summary they accounted for all of the major irregularities of the Moon 
in longitude and latitude with an accuracy not surpassed for about 
2,500 years. It's basic premises are still in use today for lunar 
calendars and cultural holidays. 
    By matching  cycles of the Moon with those of the Sun they 
developed a remarkable eclipse scheme. They could assess the chance of 
seeing a predicted eclipse based on the position of the Sun and Moon 
relative to the nodal points. They suffered from want of eclipse data 
from remote places around the world. They were limited to what was 
observed from the Middle East, plus haphazard reports from travellers. 
    Lunar observations were taken relative to ecliptic Normal Stars, 
conjunctions with planets, and moments of full and new Moons. With no 
trigonometry, they worked out latitude and longitude by sketching 
triangles of ecliptic-star-Moon. 
    By scaling off of the diagram, knowing the lat-lon of the star and 
the angle & distance of the Moon from it, the Moon's ecliptic coords 
were derived. This same read-from-sketch technique is used today as a 
camping & hiking  skill. It gives amazingly good results without heavy 
    The instants of full and new Moons were calculated from the 
relative rising of the Moon and Sun. Since full Moon will rarely occur 
exactly at sunset, the Moon rises a little before or after sunset. In 
the former case the Moon is a few hours before exactly full; the 
latter, a few hours after. 
    By proportion of the intervals of the Moon-Sun risings, the moment 
of full Moon was found. A similar logic applied to the Moon-Sun 
settings. A parallel construction was applied to the risings and 
settings of the Moon and Sun near new Moon. 
    To cover for adverse weather, the Babylonians used tables of 
previous observations to fill in an observation lost to clouds. An 
amazing feature of these tables is that they fully considered the 
varying speed of the Moon around the Earth. This was never part of an 
orbit concept and there is nothing of a perigee/apogee idea. It just 
was a feature of the Moon that she at times ran faster or slower than 
the mean speed. 
    This, by current model, is due to the Kepler effect in her 
elliptical orbit. Babylonians simply noted that at opposite sides of 
the ecliptic the Moon ran a bit faster or slower than average. What's 
more, these opposite points migrated around the zodiac at the rate we 
know today for the line of apsides. 
    They even found the variation in speed now known to be caused by 
the Sun's perturbation. To them the Moon moved a bit slower from new 
to full and faster from full to new. Not by much but enough to be 
detected with their obsrevational skills. 
Indian influence 
    During the breaks and lunch one topic bantered about was the 
connection between the astronomies of India and Mesopotamia. 'India' 
and 'Indian' here means the region now occupied by northern northern 
India and Pakistan. 
    From these discussions, it seems that Indian and Mesopotamian 
astronomy developed separately from each other. The two were isolated, 
save for occasional exchanges incidential to travels, until the 
Persian conquest of Babylonia. Persia had a more routine interaction 
with india and then carried over some of its astronomy into Babylonia 
in the mid hundreds BC. 
    It also seems that we have texts from India that show a mature 
astronomy from several hundred to a thousand years before the oldest 
good cuneiform texts. From this it seems that a sophisticated 
astronomy flourished in India before that of Babylonia got going.
    Indian astronomy didn't percolate to the West until the present 
era. The most obvious legacies are the decimal number system, the 
zero, and 'Arabic' numeric characters.
    Mesopotamia reached an amazingly high level of mathematics for 
both practical work and pure exercise. They worked out problems of 
plane and solid geometry, algebra, approximate spherics, simultaneous 
equations, a crude calculus. 
    They had nothing of trigonometry altho they dealt with situations 
normally calling for it. Their methods were approximate yet fully 
adequate for their needs. In many cases they solved their problems by 
iteration and logic. 
    They published tables of multiples, powers, roots, reciprocals. 
Division was for them as difficult as it is for us today. Their work-
around was ours before calculettes. To divide by a number, multiply by 
its reciprocal. The value was looked up in a book of reciprocals. The 
same process, in Mesopotamia and until very recent times, was done for 
powers and roots. 
    Irrational numbers were handled by approximate ratios. Pi was 22/7, 
just as it is commonly treated today. Other close fractions were used, 
all yielding thoroly good results. 
    They knew and used the Pythagoras triangle of 3-4-5 and 5-12-13. 
They realized that these two were exact triangles with integer sides. 
All other right triangles had an irrational hypotenuse. 
    They also knew that a square had an irrational diagonal, sqrt(2) 
times the side. They found that an almost-square of sides 20 and 21 
had a diagonal of exactly 29. This was handy for surveying and 
building structures. 
    Other geometry included areas and volumes of various shapes, 
converting one shape into an other of same area, finding area of a 
missing part of a larger shape. There were few purely angular methods 
like what is the nagle between two sides of a figure. In most cases 
angles were cited as rise-over-run. 
    In algebra Babylonia worked out equations of second degree and had 
a general solution for a quadratic equation. More complex equations 
were solved by iteration. 
   They applied crude, but effective, techniques of integration and 
differentiation. The Mesopotamians developed tables of first and 
second differences, much as we did before electronic calculettes. 
    The Neugebauer conference and 'Before Pythagoras' exhibit were one 
of the marvelous features of astronomy offered in the City. It was 
litterally a free two-day course, plus repeat visits, in ancient 
Mesoptoamian maths and astronomy. It was staged in a vigorous district 
of Yorkville with eating and shopping during lunch and after meeting 
hours. It was convenient to other parts of the City by express trains 
at the 86th Street station on the Lexington Avenue line or by buses in 
the busier streets. 
    This first-hand experience with the tablets in the exhibit and the 
slides in the presentations helps fortify the legacy in the home 
astronomer that he walks in a profession of at least four thousand 
year endurance. When he examines a current almanac or ephemeris, he 
sees the very same events and features of the planets and Moon and 
eclipses first described by his ancestors of that far ago era in the 
clay fields of Mesopotamia. 
    This can only be further strengthened by viewing the upcoming 
lunar eclipse on 21 December 2010. Is that  a Babylonian skywatcher 
next to you, his hand on your shoulder, explaining the event?