Combining Science and History: Lis Brack-Bernsen’s Contributions to the History of Babylonian Astronomy

Introduction

Lis Brack-Bernsen is a historian of ancient astronomy and mathematics who for many years conducted research and taught in the history of science program at Regensburg University in Germany. Now retired from the university, she remains very active in research into the history of Babylonian astronomy and astrology. She is the author of two books, author or coauthor of more than 35 research papers published in journals and edited collections, and editor of one coedited collection. For many years she served on the editorial board of the history of science journal Centaurus and was founder of the “Regensburg” series of workshops on the history of Babylonian astronomy. Her work, which is characterized by detailed mathematical and astronomical analysis alongside careful study of the original source material, has transformed the way that we understand the relationship between ancient Babylonian mathematical astronomy and its empirical basis and along the way revealed a whole new side of Babylonian astronomy that no one knew existed. In 2017, Brack-Bernsen’s colleagues presented her with a Festschrift in her honor 21.

Education and Training in the History of Science

Born and brought up in Denmark, Brack-Bernsen studied mathematics, physics, and astronomy at the University of Copenhagen. As a student, she also took courses in the history of ancient mathematics and astronomy taught by Olaf Schmidt, who profoundly influenced Brack-Bernsen’s approach to the study of ancient science. Schmidt had himself graduated from the University of Copenhagen in 1938 with a degree in mathematics, and, just as Brack-Bernsen had been introduced to the history of ancient astronomy and mathematics by him, he had been introduced to the subject by Otto Neugebauer, unquestionably the most influential scholar of the history of ancient exact science during the twentieth century. In 1939, Neugebauer left Europe and took up a professorship in the mathematics department of Brown University. Schmidt followed Neugebauer to the USA and was appointed an instructor in mathematics and a research assistant to Neugebauer at Brown. Because of the German occupation of Denmark, Schmidt was forced to stay at Brown until the end of the war. Although personally difficult for Schmidt, his time at Brown was scientifically very fruitful. In 1943, he was awarded a PhD for his thesis “On the Relation between Ancient Mathematics and Spherical Astronomy,” in which he demonstrated the importance of the observation of phenomena on the horizon and their role in the problem of determining the arc of the celestial equator which rises or sets in the same time as a given arc of the ecliptic 6.

In 1953, Schmidt was appointed a member of the Mathematical Institute of the University of Copenhagen, and it was here, a decade later, that his and Brack-Bernsen’s paths crossed. As Brack-Bernsen recalled in her obituary notice for Schmidt:

Olaf Schmidt was a careful, friendly, and very pedagogical teacher, and a brilliant and sharp thinker. He was also modest and unafraid of explaining things thoroughly. He admired and respected the ancient scientists, analysing their problems and results by means of modern science and, at the same time, trying to understand them on their own terms. 6, p. 132

The last point is crucial: Schmidt saw modern science as a tool to help analyze ancient scientific material, but he was always careful to ensure that ancient science was not interpreted as the same as modern science, nor “translated” into its modern equivalent. Brack-Bernsen explains:

Since our modern mathematical notation implies much knowledge which was unknown to the ancient mathematicians, he [Schmidt] never used modern notation in his lectures but invented a notation that came as close as possible to ancient mathematical thought. In his own words: “I want my representation to be such that an imaginative ancient Egyptian or Greek sitting in the back row of this room and listening to my lectures would nod his head as if saying: ‘yes, that is right, that is fairly what I meant when I wrote it.’” 6, p. 132

Schmidt’s research and teaching covered the history of astronomy and mathematics in Greece, Egypt, and Babylonia. His classes led directly to some of Brack-Bernsen’s own work, though, as she told me, Schmidt always wanted his pupils to publish alone, without his name. Only occasionally would he reluctantly consent to be named as a coauthor. As Brack-Bernsen recalls:

I had to force him to publish the paper “On the Foundations of the Babylonian Column Phi” together with me. Section 7 was his part of the paper. It was what he had taught in his lectures on Babylonian astronomy—and based on this knowledge I had a deeper understanding of what was going on. But still he did not want to be second author. 1

Schmidt’s influence on Brack-Bernsen’s work can be seen in other, less direct, ways too. Like Schmidt, Brack-Bernsen has emphasised the importance of horizon observations in ancient astronomy and has successfully and carefully used modern mathematical methods and synthetic astronomical data to investigate ancient mathematical astronomy, but has always taken care to ground these investigations in the original source texts and to explain these texts as much as possible from a Babylonian, not a modern, perspective.

Schmidt was influential in the development of Brack-Bernsen’s career in other ways too. In 1969, while still a student at the University of Copenhagen, at Schmidt’s invitation she published her first paper in volume 14 of the journal Centaurus, a special issue in honor of Otto Neugebauer’s retirement. The paper “On the Construction Column B in System A of the Astronomical Cuneiform Texts” was written in response to a question posed to Brack-Bernsen by Schmidt about how the monthly change in the position of the moon at conjunction and opposition is influenced by solar and lunar velocity. Brack-Bernsen calculated the moon’s position for more than 223 consecutive conjunctions and oppositions by linear interpolation from data found in the Nautical Almanac. She plotted the monthly change in lunar position and showed that the dominant effect was the solar anomaly: the graphs show an overall sinusoidal form with a period of the solar year. Working by hand, she then plotted yearly curves of the same data, with each year drawn with a different color pencil, and found that the variation of the curves up and down was due to lunar anomaly and had a period of 223 months, a period known as the Saros. As I have seen firsthand, this approach of drawing curves in different colors to identify the less-dominant contribution to a function’s variation has been instrumental in several of Brack-Bernsen’s later studies of the interplay between lunar and solar anomaly.

Following his retirement, Brack-Bernsen would visit Schmidt at home to discuss her ongoing research. Brack-Bernsen has also inherited her teacher’s generous nature as a scholar, encouraging and supporting the work of others, including myself, and providing kind but crucial comments at critical stages in their research. As she wrote in Schmidt’s obituary:

I myself feel deeply indebted to my teacher whom I have the privilege to know for almost 30 years. Through his lectures and the interesting problems he posed to me, he introduced me to the exciting domain of the history of astronomy and transmitted his deep insight into the questions and working methods of ancient mathematics and astronomy. I gratefully remember countless occasions at his home, where I could also discuss my newest results with him. 6, p. 133

Babylonian Astronomy

Roughly five thousand cuneiform tablets recovered from the site of Babylon and other ancient cities in what is now Iraq provide evidence for the practice of astronomy and astrology 20. These texts mostly date to the middle and end of the first millennium BC but it is apparent that some form of astronomical activity already existed in the second and perhaps even the third millennium BC. Babylonian astronomy in the latter half of the first millennium BC encompassed a wide range of activities including the regular and systematic observation of selected astronomical phenomena, the development and use of methods to compute future astronomical phenomena, and the astrological interpretation of astronomical data.

The astronomical phenomena of interest to the Babylonian scholars in the first millennium BC were predominantly the regular or cyclical phenomena exhibited by the five planets visible to the naked eye (Mercury, Venus, Mars, Jupiter, and Saturn) and the moon. For the planets, these phenomena are the dates of the passages of a planet past a group of reference stars in the zodiacal band (called Normal Stars in modern scholarship) or past another planet together with the distances of the planet above and below the star or planet on that date, and the dates and positions of a planet when it exhibits one of its synodic phenomena (first and last appearance, station, or acronychal rising). For the moon, the regular phenomena are the dates of the passages of the moon past those same Normal Stars or planets with the distances of the moon from the other body, lunar and solar eclipses, and a group of six time intervals between the sun and moon crossing the horizon on specific occasions each month. These six intervals, collectively known as the “lunar six” in modern scholarship, have been a particular focus of Brack-Bernsen’s scholarship and so it is worth explaining them in detail here.

In the Babylonian calendar, the month was tied to the cycle of the moon. The month began on the (observed or computed) evening of the first appearance of the new moon crescent shortly after sunset, an event that occurs a day or two after conjunction of the sun and moon. Due to the variability in the length of the time interval between one conjunction and the next (this interval is called the synodic month) caused by the varying speeds of the sun and moon in their apparent orbits as we see them around the Earth, the Babylonian month can have either twenty-nine or thirty days. Full moon can occur on the thirteenth, fourteenth, fifteenth, or sixteenth day. The lunar six time intervals were measured on the evening of the first day of the month, on the evening and mornings around full moon, and on the morning of the last visibility of the moon before conjunction. The intervals are listed below (confusingly, the Babylonians named two of these intervals “NA,” and so I will use NA_new and NA_full to distinguish them):

NA_new: The time between sunset and moonset when the moon was visible for the first time after conjunction (i.e., on the first day of the month).

ŠÚ: The time between moonset and sunrise when the moon set for the last time before sunrise.

NA_full: The time between sunrise and moonset when the moon set for the first time after sunrise.

ME: The time between moonrise and sunset when the moon rose for the last time before sunset.

GE₆: The time between sunset and moonrise when the moon rose for the first time after sunset.

KUR: The time between moonrise and sunrise when the moon was visible for the last time before conjunction.

The size of these intervals, the days of the month on which they occur, and even the order in which ŠÚ, NA_full, ME, and GE₆ take place are very complicated to determine in advance because they depend upon several factors including the variable speed of the sun and moon (solar and lunar anomaly), the moon’s latitude at the conjunction or opposition, the angle between the ecliptic and the horizon, and the time of conjunction or opposition relative to sunrise and sunset. The lunar six time intervals themselves were measured to a precision of 10 NINDA or 1/6th of an UŠ, where one UŠ is equivalent to four of our minutes and so 360 UŠ equals a whole day of 24 hours. The time intervals were written using the sexagesimal number system where each “digit” is sixty times greater than the “digit” to its right. It is conventional to transcribe sexagesimal numbers with commas separating each “digit” and a semicolon separating integers from fractions.

Regular, night-by-night, reports of astronomical observations were recorded in a group of texts called “regular watching” by their authors or “Astronomical Diaries” by modern scholars. These texts typically cover half a year and are divided into sections for each month. Most of each section is taken up with daily accounts of astronomical phenomena. At the end of each section we find summary statements of the positions of the planets during the month, the value of staple commodities in the market at Babylon, the river level as the Euphrates flows through the city, and selected historical events. Although the main aim of the Diaries was to record observed astronomical phenomena, bad weather often prevented observations being made. On these occasions, a prediction of the phenomenon was often recorded instead. Predicted phenomena could include the lunar six intervals, eclipses, and planetary synodic phenomena.

The Diaries provided the source material for various texts containing compilations of astronomical data. Of particular interest are the so-called “Goal-Year Texts” which gather together data which are to be used in making predictions of the astronomical phenomena which will occur in a given year. These texts exploit the fact that there are certain periods, different for each planet, after which phenomena will recur on more or less the same day in the Babylonian calendar. For example, phenomena of Venus recur nearly precisely after eight Babylonian years. Thus, predictions of the phenomena of Venus for a coming year can be produced by simply going back to the observations made eight years previously (in practice, a few small corrections were applied to make the predictions slightly better) 19. The Goal-Year Texts also contain data for eclipses and the lunar six. However, the lunar six, in particular, do not repeat exactly after one period and scholars were for a long time baffled as to how they Babylonians predicted them.

By at latest the end of fourth century BC, the Babylonians had also developed a second type of predictive astronomy which calculated lunar and planetary phenomena using purely mathematical methods which did not rely upon the direct use of previous astronomical observations 1718. Instead, in this mathematical astronomy the dates and celestial longitudes of the moon at new and full moon or of a planet when it exhibits one of its synodic phenomena were calculated using arithmetic procedures. These procedures were used to produce tables of computed lunar or planetary phenomena. The entries in each row of the table are computed either directly from those in the row above or else by adding values found in earlier columns of the current row. The planetary systems are relatively simple and apply one of two mathematical functions known as step functions and zigzag functions to the date and position of the planet when it exhibits a particular phenomenon to calculate the date and position when it next exhibits the same phenomenon. Step functions assign a particular value that is to be added to the previous date or position according to the position of the planet in the zodiac. Zigzag functions increase and decrease by a constant amount between fixed maximum and minimum values and determine the value which is to be added to the value in the previous row of a table. As I will discuss below, the lunar systems are both more ambitious in what they calculate and more complicated in how they do so than the planetary systems.

Astronomy had many roles in Babylonian society. Two of its main uses were in regulating the calendar and in providing data to be used in astrology. As mentioned earlier, the Babylonian calendar used lunar months which can least for either 29 or 30 days. The sequence of 29- and 30-day months, however, is extremely irregular: although, on average, more or less the same number of 29- and 30-day months will occur within a given year, these months are unevenly distributed. For example, while sometimes 29- and 30-day months may alternate for several months, on other occasions, three or even four months of one kind may occur in a row. This irregularity in the length of the month is caused by the large number of astronomical factors which influence the visibility of the new moon crescent including the variable speeds of the sun and moon (solar and lunar anomaly), the variation in the moon’s latitude, and the variation of the inclination of the ecliptic to the horizon over the course of the year, not to mention the observing conditions (in particular due to weather). In addition, twelve 29- and 30-day months fall somewhat more than ten days short of the length of the solar year. In order to keep the months in line with the seasons, it is necessary to add an extra “intercalary” or “leap” month roughly once every three years. Determining in advance when a month begins and how many months there are in the year are therefore complicated astronomical questions. Babylonian astrology during the late first millennium BC also became increasingly mathematical, relying upon calculated data and mathematical schemes drawn from astronomy.

Lis Brack-Bernsen’s work has spanned all aspects of Babylonian astronomy, including the mathematical schemes found in one of the earliest astronomical texts known as MUL.APIN and the mathematics and astronomy underlying some late first millennium BC astrological schemes 914. In the following, I will focus on two key areas in which her work has had a particularly large impact on our understanding of the history of Babylonian astronomy.

The Origin of Column $\Phi$ of Lunar System A

Lunar System A aims at calculating the time of conjunction and opposition of the sun and moon, the moon’s longitude at that moment, the visibility of the moon around conjunction and opposition, in particular the visibility of the new moon crescent and therefore the length of the lunar month, and whether or not an eclipse of the sun or moon will happen at that conjunction or opposition and, if so, the magnitude of the eclipse. The moon’s longitude at conjunction and opposition is computed by a simple two-zone step function. Calculation of the time of conjunction or opposition, the visibility of the moon, and whether or not there will be an eclipse is much more complicated, however. For example, the time of conjunction or opposition depends upon both solar and lunar anomaly (i.e., the variable speeds of the sun and the moon), and therefore cannot be modelled by a simple mathematical function. Instead, the Babylonians developed a method of separating the effects of the two anomalies, modelling them individually, and then combining their contributions to calculate the time of conjunction or opposition. Similarly, the determination of whether or not there will be an eclipse involved modelling the moon’s motion in latitude by combining functions for the moon’s longitude with one for the motion of the moon’s node, and then using a further calculation to derive the magnitude for the eclipse. If the magnitude falls within certain limits then an eclipse is predicted. Otherwise, there will not be an eclipse.

The internal mathematical structure and the astronomical meaning of the various functions within the lunar systems (usually referred to as “columns” because they appear as such in the tables) was placed on a firm footing by Otto Neugebauer’s detailed 1955 study Astronomical Cuneiform Texts 17. However, Neugebauer did not address the question of the empirical basis of the functions or how they were constructed. It is this question that has been the focus of much of Brack-Bernsen’s work.

At the heart of lunar System A lies column $\Phi$. Column $\Phi$ is a linear zigzag function with minimum 1,56,47,57,46,40, maximum 2,17,4,48,53,20, and line difference 2,45,55,33,20. Column $\Phi$ has a number period of 6248, meaning that it takes 6248 lines before we reach the same number in the zigzag function. Over that period, values have gone up and down the zigzag 448 times. 6248 lines corresponds to 6248 lunar months or just over 505 years. In the lunar tables, column $\Phi$ is the first column calculated, appearing right after the date column. Within the tables, $\Phi$ serves purely to generate two other columns: column F, which represents the daily displacement of the moon on the day of syzygy (i.e., the lunar velocity), and column G, which is used in the determination of the length of the synodic month. The length of the synodic month is determined by adding together columns G and J to give the excess length of the month over 29 days. These two columns represent contributions due to lunar and solar anomaly respectively. Thus, column $\Phi$ is used to generate two functions, column F and column G, both of which are directly related to lunar anomaly.

Within a System A lunar table, column $\Phi$ is used purely as a generating function. However, some prose texts which set out calculations using column $\Phi$ and other functions, as well as some other, nonstandard, tabular material, show that column $\Phi$ itself represents the lunar contribution to the length of 223 synodic months, a period known today as the Saros after which there is a close return of both lunar anomaly and the latitudinal motion of the moon (as a consequence, the Saros is an eclipse period). Like the length of the month, the length of the Saros depends upon lunar and solar anomaly. Column $\Phi$ is the lunar contribution to the length of the Saros in excess of 6585 whole days; this contribution would need to be complimented by a solar contribution, although no such function has yet been found in the preserved texts. Column $\Phi$, the lunar contribution to the length of the Saros, and column G, the lunar contribution to the length of the synodic month, are both normed for maximum solar anomaly. In other words, the solar correction is zero when the solar anomaly is greatest, and negative for smaller values of the solar anomaly.

Since column $\Phi$ is connected to the length of the Saros, it might be expected that it could simply be derived for measurements of the time between two eclipses separated by 223 months. Here is the rub, however: Column $\Phi$ represents only the contribution of lunar anomaly to the length of the Saros but by modelling the variation in the length of the Saros using modern data Brack-Bernsen showed that the variation in the length of the Saros is dominated by the effect of solar anomaly—lunar anomaly plays a much smaller role 3. Thus, simple observations of the length of the Saros cannot have been used to derive column $\Phi$. How was it derived, then?

Brack-Bernsen’s answer to this question brings us back to the lunar six time intervals discussed in the previous section. Recall that the lunar six intervals are the time between the moon and the sun crossing the horizon. Individually, the lunar six intervals are highly complex functions, depending upon several factors including the lunar velocity, the inclination of the ecliptic to the horizon, the moon’s latitude and the time of conjunction or opposition relative to sunrise and sunset. However, as Brack-Bernsen has shown, several of these factors cancel out when some of the lunar six are added together 14. In particular, if we add the four intervals around full moon (ŠÚ, NA_full, ME, and GE₆), which together, following Brack-Bernsen, we call the “lunar four,” all of the factors except for lunar velocity are eliminated (at least to a first approximation). Thus, the sum of the lunar four varies only due to lunar anomaly.

Through a systematic analysis of a large body of synthetic “observations” of the lunar four, Brack-Bernsen also discovered that the sum of the lunar four not only varies according to lunar anomaly, and thus is in phase with column $\Phi$ at full moon, but also that the amplitude of variation of the sum of the lunar four is very close to that of column $\Phi$ 457. The values of column $\Phi$ are offset from the sum of the lunar four by about 100 UŠ; Brack-Bernsen has recently suggested a plausible explanation for how the Babylonians made this 100 UŠ adjustment to obtain column $\Phi$ drawing upon Babylonian methods of determining the time of eclipses 10.

Brack-Bernsen’s groundbreaking discovery of the astronomical information embedded within the lunar six time intervals and how the individual intervals can be combined in order to provide information on lunar anomaly has opened up the doorway to understanding the empirical origin of column $\Phi$. Whether or not it is correct in all of its details, which remains to be seen, it certainly takes us a good part of the way there. Indeed, important aspects of her proposal have been used by other scholars proposing their own (and, in my opinion, less likely) reconstructions of the development of column $\Phi$ 1516.

The Prediction of the Lunar Six Intervals and the Length of the Month

Brack-Bernsen’s careful investigation into the astronomical meaning of the lunar four time intervals and their partial sums also led to the remarkable and totally unexpected discovery of a simple, elegant, yet very accurate method developed by the Babylonians to predict these intervals. This method draws on the fact that, although the individual lunar four are dependent upon many factors, most of these factors are eliminated or at least minimized if we take the sums ŠÚ+NA_full and ME+GE₆. Indeed, ŠÚ+NA_full corresponds to the setting time of the arc of the ecliptic between the moon’s position when it sets on the two consecutive mornings around full moon, and ME+GE₆ is the rising time of the arc of the ecliptic between the moon’s position when it rises on the two consecutive evenings around full moon. The setting time and the rising time correspond to the projection of a portion of the ecliptic onto the celestial equator. The two intervals therefore respectively correspond to the stretch of the celestial equator which rises or sets in the equivalent to one day’s worth of the moon’s motion.

Brack-Bernsen further demonstrated that many of the effects upon the size of the lunar four intervals repeat after 1 Saros of 223 synodic months. However, 1 Saros is equal to approximately 6585 + 1/3 of a day. Because of this 1/3rd of a day, the value of the lunar four time interval after 1 Saros will vary by the rising or setting time corresponding to 1/3rd of a day’s worth of lunar motion. And since the equivalent value for one whole day is equal to ŠÚ+NA_full and ME+GE₆, then the values for 1/3rd of a day will be 1/3rd of ŠÚ+NA_full and 1/3rd of ME+GE₆, depending upon whether we are dealing with the lunar four which take place in the morning or the evening. This can be expressed by the following four equations, where n corresponds to the month we wish to compute for 8.

$$\begin{equation*} \begin{aligned} \text{ŠÚ}(n) &= \text{ŠÚ}(n\text{–}223) + 1/3(\text{ŠÚ}(n\text{–}223) +\text{NA}_{\text{full}}(n\text{–}223)) \\ \text{NA}_{\text{full}}(n) &= \text{NA}_{\text{full}}(n\text{–}223) \text{–} 1/3(\text{ŠÚ}(n\text{–}223) +\text{NA}_{\text{full}}(n\text{–}223)) \\ \text{ME}(n) &= \text{ME}(n\text{–}223) + 1/3(\text{ME}(n\text{–}223) +\text{GE}_6(n\text{–}223))\\ \text{GE}_6(n) &= \text{GE}_6(n\text{–}223) \text{–} 1/3(\text{ME}(n\text{–}223) +\text{GE}_6(n\text{–}223)) \end{aligned} \end{equation*}$$

Brack-Bernsen further showed that because the situation at the eastern horizon (i.e., at sunrise) is identical to that at the western horizon (i.e., at sunset) half a year later and vice versa, these rules can be expanded to the computation of the lunar six intervals at new moon:

$$\begin{equation*} \begin{aligned} \text{NA}_{\text{new}}(n) &= \text{NA}_{\text{new}}(n\text{–}223) \text{–} 1/3(\text{ŠÚ}(n\text{–}229) +\text{NA}_{\text{full}}(n\text{–}229))\\ \text{KUR}(n)&= \text{KUR}(n\text{–}223) + 1/3(\text{ME}(n\text{–}229) +\text{GE}_6(n\text{–}229)) \end{aligned} \end{equation*}$$

Having discovered these methods of predicting the lunar six, Brack-Bernsen then found direct textual evidence that the Babylonians knew them. The text TU 11 was identified as containing astronomical rules and a drawing of the tablet published already in 1922. However, despite attempts by several scholars, no one had succeeded in interpreting the text until Brack-Bernsen realized that part of it contained rules for computing the lunar six which were equivalent to the equations given above. Working with the Assyriologist and expert in translating Babylonian astronomical texts Hermann Hunger, Brack-Bernsen undertook a detailed investigation of this and related tablets 1112, recovering these rules for the lunar six and rules for determining whether or not a month had 29 or 30 days. I quote below one part of one of these texts describing the procedure for computing the value of NA_new and whether the preceding month had 29 or 30 days:

In order for you to calculate (whether the month is) full or hollow, you hold in your hand your year, you return 18 years behind you, and in the 18th year[…] you return 6 months behind you, and you take ŠÚ and NA of Month VII and add (them) together, and [you take one third of it, and] you subtract it from the NA of the 1st day of Month I of the 18th year, and whatever[…] when it[…] if it is less than 10 UŠ, predict (the month) as full. 12

The tablet on which this text is written is damaged (missing passages of text are indicated by […]). But we can see that the text tells us to take the value of NA_new from 18 years (i.e., 1 Saros or 223 months) earlier, then take the values of ŠÚ and NA_full from a further 6 months earlier, add these latter two together and take one third of the result, and then subtract it from the value of NA_new from 18 years earlier. This procedure is exactly the same as the rule given in the form of an equation above. The following text is damaged but from related texts we know that it was stating that if the value of the NA_new that has been computed is smaller than 10 UŠ, the new moon crescent is too small to be seen and so it is necessary to add the whole of ŠÚ and NA_full to it to get the value of NA_new for the next day when the moon will be seen for certain.

Babylonian texts provide evidence that this procedure for computing the value of NA_new was regularly used in practice. Let us consider the following example. The text ADART VII 30 is a Normal Star Almanac containing predicted astronomical data for the year SE 96. The section for Month IX gives 19;50 UŠ for the predicted value of NA_new that month. In order to make the prediction, we need the value of the NA_new from 223 months earlier and the sum of ŠÚ+NA_full from 229 months earlier. These values are preserved in the Goal-Year Text ADART VI 10:

$$\begin{align*} &\text{$\mathrm{NA}_{\mathrm{new}}$ for SE 78 Month IX} = \text{13;0 UŠ}\\ &\text{ŠÚ for SE 78 Month III} = \text{4;50 UŠ}\\ &\text{$\mathrm{NA}_{\mathrm{full}}$ for SE 78 Month III} = \text{5;30 UŠ} \end{align*}$$

Following the procedure as described in the text quoted above, we start by adding ŠÚ and NA_full for SE 78 Month III:

$$\begin{equation*} \text{ŠÚ} + \mathrm{NA}_{\text{full}} = \text{4;50 UŠ} + \text{5;30 UŠ} = \text{10;20 UŠ} \end{equation*}$$

Next, we take one-third of ŠÚ+NA_full and round to the nearest 0;10:

$$\begin{equation*} 1/3(\text{ŠÚ}+ \mathrm{NA}_\mathrm{full}) = 1/3 \times \text{10;20 UŠ} = \text{3;30 UŠ} \end{equation*}$$

We subtract this 1/3(ŠÚ+NA_full) from the NA_new for SE 78 Month IX to give the value of NA_new for SE 96 Month IX:

$$\begin{equation*} \mathrm{NA}_{\text{new}} = \text{13;0 UŠ} - \text{3;30 UŠ} = \text{9;30 UŠ} \end{equation*}$$

However, 9;30 UŠ is below the visibility limit of 10;0 UŠ for NA_new and so we must add the whole of ŠÚ+NA_full to this result:

$$\begin{equation*} \mathrm{NA}_{\mathrm{new,corrected}} = \text{9;30 UŠ} + \text{10;20 UŠ} = \text{19;50 UŠ} \end{equation*}$$

This result, 19;50 UŠ, is exactly the value given in the Normal Star Almanac.

The discovery of these methods for predicting the lunar six values has transformed our understanding of Babylonian astronomy.

Final Thoughts

Lis Brack-Bernsen’s work on the history of ancient Babylonian astronomy serves as a model for how the detailed investigation of ancient astronomical theories and methods, with judicious and careful use of modern mathematics and astronomy as tools, can yield important insights into ancient scientific practice. Behind much of her work has been an interest in exploring the empirical basis for the Babylonian methods of astronomical computation. Through the surviving Astronomical Diaries and related texts we know what phenomena were regularly observed by the Babylonians and with what precision positions and time intervals were recorded. The preserved records are fragmentary, however, and as such not suited to large-scale analysis. In much of her work, therefore, Brack-Bernsen has turned instead to simulated observations, which can be analyzed both systematically and on large scales. In particular, she has made extensive use of the computer code “AA” developed by S. Moshier to simulate Babylonian lunar six data. Through her analysis of this data, Brack-Bernsen was able to find and confirm the correctness of empirical rules for calculating lunar phenomena. Sometimes these rules are hinted at in texts, but not fully explained. For example, the Goal-Year Texts indicated that the partial sums of the lunar four for eighteen and a half years before the target “goal” year were calculated, but offered no indication of how they were to be used. Through her analysis of a large body of synthetic data of these partial sums, Brack-Bernsen was able to identify the astronomical significance of the sums and ways that they could be used in making predictions of future phenomena, and was subsequently able to find these rules written in texts that had hitherto resisted interpretation. This approach also underlies her work in reconstructing column $\Phi$ of lunar System A.

Combining this scientific type of analysis with a commitment to paying close attention to the ancient evidence itself and to trying to understand ancient astronomy through the eyes and the methods of the ancient astronomers themselves, as Brack-Bernsen has done throughout her career, brings us as close as possible to understanding ancient astronomy. Her work demonstrates that in the history of science it is essential to consider both the history and the science. Without one or the other, we can only gain a partial, and often misleading picture of ancient scientific practice. But by carefully combining both historical and scientific approaches, we can gain a deeper insight into ancient science itself.