# Predicting from Nilometer Data in Ancient Egypt: A Source of Interest for Statistical Learning in Sustainability Governance

Cyprien Gilet

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# Predicting from Nilometer Data in Ancient Egypt:
A Source of Interest for Statistical Learning in
Sustainability Governance

Cyprien Gilet

Université de Technologie de Compiègne

Heudiasyc Laboratory, UMR CNRS 7253

Compiègne, France

cyprien.gilet@hds.utc.fr

***Abstract***—With the emergence of artificial intelligence in sustainability governance, statistical and machine learning algorithms play an important role in prediction and forecasting tasks, whether in environmental, social and economic dimensions. These approaches are often seen as very promising but also raise concerns regarding trust, ethics, and the role of humans in these models. In order to provide a complementary perspective face to these questions, a relevant and important approach is to look at the past in order to understand how older similar instruments were used, their benefits and the aspects that needed to be improved. While the field of statistical and machine learning algorithms is quite recent, the paradigm of prediction based on observed data existed long before the emergence of statistical learning theory and certain data-based predictive instruments like Nilometers in ancient Egypt have already played a major role in sustainable governance for millennia. In order to provide a complementary reflection regarding these two aspects, and based on several different sources whether in the fields of archaeology, history of mathematics, accounting or geography, the objective of this paper is to present a survey gathering the recovered pieces of information regarding Nilometers in ancient Egypt and their utility as data-based predictive instruments. This paper aims moreover to highlight the reflection that Nilometers can be viewed as an ancient root of predictive models from data (including in statistical learning) and that their longevity and sustainable engineering remain sources of interest for the field of sustainability governance.

***Index Terms***—Ancient Egypt, Nilometers, sustainability governance, predicting from data, statistical learning.

## I. INTRODUCTION

Sustainability governance refers to the systems and processes that guide organizations to integrate sustainability into their management with respect to both environmental, social and economic dimensions. In recent years, Artificial Intelligence (AI) has been more and more referred as a hopeful tool for this purpose [1]–[3], but also raise concerns regarding trust, ethics, and the role of humans in these models [1]. For instance, according to [1], AI models can enable more than 130 targets across the Sustainable Development Goals (SDG) but may also inhibit more than 50 other ones, both in short

and long terms, regarding both environmental, societal and economic outcomes.

Among the commonly reported positives impacts, AI models can for instance support the improving of quality education, sustainable cities, water quality and sanitation, precision medicine, climate change understanding and adaptation, adaptative agriculture face to climate change, provision of food, health of ecosystems and low-carbon energy systems. However, regarding the commonly reported negative impacts, the growth of AI models could also lead to economic inequalities within nations due to unequal available ressources. These models also consume a lot of energy in their use. They also raise ethics considerations. For instance, predictive models can suffer from unfair predictions, lack of robustness or lack of transparency.

In order to increase the benefits that AI could bring regarding Sustainable Development Goals and to reduce the negative impacts, two major and complementary aspects would be beneficial to consider:

1. How can we improve our present models to address the previously reported issues? This question can be addressed by present and future research, not only in the field of AI or machine/statistical learning, but also in collaboration with other fields like for instance environment, economy, society, philosophy, medicine, or urbanism.

2. Looking at the past, before the emergence of AI, in order to understand how we used to face older challenges related to both environmental, societal and economic dimensions. How older civilisations managed to overcome previous climate changes? How older civilisations managed to be fair regarding food distribution, regarding the economic outcomes of nations, regarding their adaptative agriculture? What worked and what needed to be improved? Can we take example regarding some aspects to improve our AI models? For instance, as mentioned in [4], bridging the gap between climate science and archaeology could bring several benefits in climate change research.
Sustainability governance using artificial intelligence is often based on Machine Learning algorithms, a field which is nowadays increasingly relevant in many application domains such as for instance medicine, ecology, economy, waste management or weather forecasting. Fitted from observed data, machine learning algorithms generally rely on statistical learning concepts [5], [6] with the aim of predicting, estimating, forecasting or understanding a variable of interest for decision making purposes.

Although the origin of machine learning algorithms emerged a little before the 1950s with the advances of computing, the foundations of learning predictive models from data are older. For instance, as mentioned in [6], statistical learning models raised up in the early 1800s with Legendre and Gauss research [7]–[9] on the method of least squares which underlies the field of regression.

To go further, the paradigm of prediction based on observed data existed even before the emergence of statistical learning. In fact, certain predictive instruments played a major role in sustainability governance long before the rise of statistical learning models. This was notably the case with Nilometers in ancient Egypt. Especially since these ancient models have played a major role in sustainability governance for millennia. Of course, these ancient models were not based on computer programs or statistical learning theory, but their longevity deserve to be emphasized and remain a root in sustainability governance. As we will recall later in this paper, Nilometers are instruments for measuring the height of the Nile [10]–[15] and in ancient Egypt, predictions about the harvest could be made according to the level of inundation of the Nile using Nilometers data [11], [12], [14]–[17] and an adaptive management could then be established for sustainability governance.

With the aim of highlighting this ancient root of predicting a variable of interest based on observed data for sustainability governance, this paper links together several important pieces of information from different sources, whether in the fields of archaeology, history of mathematics, accounting or geography, drawing parallels with the notions of our era in statistical learning. More precisely, the objectives of this paper are as follows:

1) To present a survey in the statistical/machine learning field gathering the recovered pieces of information regarding Nilometers in ancient Egypt and their utility as data-based predictive instruments, especially for sustainability governance.

2) To highlight a reflection regarding the fact that Nilometers can be considered as a root of statistical learning models and to highlight which pieces of information would be interesting to recover in the fields of mathematics in ancient Egypt and archaeology in order to complement and confirm this discussion.

3) To present a complementary and historical source of reflection regarding the philosophical, ethical and sustainability questions of our era related to data-based predictive models for sustainability governance.

The paper is organized as follows: Section II reminds the paradigm of predicting based on observed data in statistical learning, Section III reminds the paradigm of predicting from Nilometer data in ancient Egypt, Section IV discusses why Nilometers can be a source of interest for statistical learning in sustainability governance, and finally Section V concludes the paper.

## II. PREDICTING FROM DATA IN STATISTICAL LEARNING

This section is devoted to recalling the core of statistical learning theory, especially in supervised learning, in order to highlight the paradigm of predicting based on observed data.

### A. Predicting based on observed data

In statistics, prediction is a part of statistical inference. Statistical learning is a framework for machine learning that addresses the statistical inference problem of finding an accurate predictive decision rule based on observed data [5], [6].

Formally, denoting $\mathcal{X}$ the $d$-dimensional feature space of attributes describing each instance and $\mathcal{Y}$ the space of predicting values, a predictive decision rule $\delta$ takes the following form

$$ \delta: \mathcal{X} \to \mathcal{Y}, \quad \text{such that} \quad \forall x \in \mathcal{X}, \delta(x) = \hat{y} \in \mathcal{Y}, \quad (1) $$

where $\hat{y}$ denotes the predicted value associated with a given instance $x$. In supervised learning, it is common to consider tow kinds of tasks depending on the type of values to predict: If the variable of interest is numeric/quantitive, the problem is called regression and if the variable of interest is categorical, the problem is called classification.

In supervised learning, the decision rule $\delta$ is fitted from a training dataset containing $n$ observed instances

$$ \mathcal{S} = \{ (x^{(i)}, y^{(i)}), i = 1, \dots, n \} \quad (2) $$

such that $x^{(i)} = [x_1^{(i)}, \dots, x_d^{(i)}]$ is a vector containing $d \ge 1$ attributes (features) describing the instance $i$ and $y^{(i)}$ characterises the observed variable of interest associated with this instance $i$. In practice, the data are usually structured in a tabular format as illustrated in Table I for which each row characterises an instance and each column characterises an attribute.

<table><caption>TABLE I
ILLUSTRATION OF COMMONLY STRUCTURED TABULAR DATA IN STATISTICAL LEARNING.</caption><thead><tr><th></th><th>Attribute 1</th><th>...</th><th>Attribute d</th><th>Variable Y</th></tr></thead><tbody><tr><td>instance 1</td><td>x<sub>1</sub><sup>(1)</sup></td><td>...</td><td>x<sub>d</sub><sup>(1)</sup></td><td>y<sup>(1)</sup></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>instance i</td><td>x<sub>1</sub><sup>(i)</sup></td><td>...</td><td>x<sub>d</sub><sup>(i)</sup></td><td>y<sup>(i)</sup></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>instance n</td><td>x<sub>1</sub><sup>(n)</sup></td><td>...</td><td>x<sub>d</sub><sup>(n)</sup></td><td>y<sup>(n)</sup></td></tr></tbody></table>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Timeline of ancient Egypt periods.</figcaption></figure>

## B. Developments in statistical learning theory

In order to address the statistical inference problem of finding an accurate predictive decision rule from the training samples (2), statistical learning models generally rely on mathematical assumptions, either regarding the statistical distribution of the data or by using mathematical modeling (e.g., linear, nonlinear models). These models are then usually fitted by minimizing an appropriate loss with respect to the context and to the training dataset (2). For this purpose, statistical learning models rely on mathematical, statistical and optimization theories, and their progress is the result of advances made in these complementary fields as well as the development of computer science.

As mentioned in [6], fitting a statistical predictive decision rule raised up in the early 1800s with Legendre and Gauss research [7], [8] on the method of least squares which underlies the field of regression. In the past century, thanks to advances in mathematics, statistics, optimization and computer science, statistical learning has known a considerable growth. Many globally recognized methods have been developed since the middle of the past century, such as Logistic Regression, Decision Trees, Support Vector Machine, and Neural Networks. Since the 2010s, deep learning algorithms have revolutionized the field of machine learning, achieving peak performances in several application areas. These algorithms can not only deal

with tabular data as in Table I, but also with images, audio, text, or video.

Alongside predictive performances, statistical learning research has also focused on the predictive confidence of these models or the management of uncertainty in data. Furthermore, ethical considerations are also extensively studied in relation to these models, whether concerning data sensitivity, the level of trust that can be placed in these models, or the sustainability of these approaches.

## C. Predicting based on observed data: The core of supervised statistical learning

While the development of supervised statistical learning models continues to grow exponentially by addressing many objectives, whether in terms of performance, robustness, confidence, ethics or sustainability, all these predictive decision rules share a common core: predicting based on observed data. This paradigm of prediction based on observed data is indeed seen as the core of statistical learning models while the common foundation is to rely on mathematical assumptions, either regarding the statistical distribution of the data or by using mathematical modelling and computer science advances.

It is however important to note that the paradigm of prediction based on observed data existed before the rise of this field. This was notably the case with Nilometers in ancient Egypt,
Map: Nile River and Historical Egyptian Sites

<figure><img src="image_3.png" /><figcaption>Fig. 2. The Nile river with fertile areas and historical sites dating back to ancient Egypt.</figcaption></figure>

instruments which have played a major role in sustainable
governance, as highlighted in the next section.

III. NILOMETERS IN ANCIENT EGYPT: AN EXAMPLE ROOT OF PREDICTIVE INSTRUMENT FOR SUSTAINABILITY GOVERNANCE

This section is devoted to highlight the paradigm of pre-
dicting based on Nilometer data for sustainability governance
in ancient Egypt, by linking together several important pieces
of information from different sources, whether in the fields of
archaeology, history of mathematics, accounting or geography.
Fig. 1 presents a timeline of ancient Egypt which may be
useful for visualizing some of the historical dates mentioned
throughout this section.

A. *The Nile in Ancient Egypt*

For millennia, the Nile has played an important role in the
agricultural and economic life of Egypt. Before the construc-
tion of the Aswan High Dam in the past century, the Nile
has provided the water needed for agriculture and irrigation,

transforming the surrounding arid lands into fertile agricultural
areas. In ancient Egypt, the agricultural year was divided into
three seasons. First, the Nile flooded the surrounding plains
every year from the late July until the end of September. With
the receding floodwaters in late October, the fields reappeared
with fertile and well-watered soils, ready for sowing. Sowing
took place after the receding (usually in November) and the
crops grew until reaching maturity between March and May,
the beginning of the third season when the harvest could finally
begin.

Crop yields depended mainly on the Nile's flood level. How-
ever, the annual flood level could vary considerably from year
to year. If the flooding was insufficient, only a small area of
cultivated land was covered by this silt, which could lead to a
risk of future famine. If the flood level was too high, this could
also become counterproductive to future crops. In order to
adapt to these uncontrollable natural flows and reduce the risk
of famine, agricultural practices evolved throughout ancient
Egypt, whether by building irrigation basins to optimize water
resources, as well as in different water supply techniques such
as *shadoufs*<fnref n="1" />, and also by measuring and recording the annual
Nile floods using Nilometers for observation and prediction.

B. *Nilometers for data acquisition*

Nilometers are instruments for measuring the height of the
Nile [10]–[15]. They can take several forms, such as series
of stairs, columns or wells. The most famous are located
on Elephantine Island in Aswan, on Roda Island in Cairo,
and in the Temples of Edfu, Kom Ombo, Philae, Esna, and
Dendera [10]–[13], [15]. Fig. 2 presents a map showing some
of these historical sites along the Nile.

Although the appearance of the first nilometers is difficult
to date, we nevertheless know that they already existed early
in Dynasty I thanks to old records of the Nile floods dating
back to 3050 B.C, during the reign of King Djer (Zer). Indeed,
among the oldest records of the Nile floods, the Palermo Stone
displays engravings dating from the Fifth Dynasty (2480 BC,
Old Kingdom) that include sixty-three flood levels dating
back to the reign of King Djer at the beginning of the First
Dynasty (around 3050 BC) [11], [12], [19]. The Palermo Stone
(illustrated in Fig. 3) is a famous engraved stele made up of
several fragments, constituting a historical and archaeological
document preserving part of the royal annals from the end
of the Predynastic period to the Fifth Dynasty of the Old
Kingdom.

As explained by R. Said in the greatly valuable book [12],
the two faces of the Palermo Stone are divided horizontally
into registers (rows) which are themselves divided vertically
into compartments, each bearing its own hieroglyphic inscrip-
tion. The uppermost row on the front lists the names of
the predynastic rulers whose reigns and actions are likely
unknown. The other rows recount the actions of the kings
of Dynasties I through V, whose names, along with those of

<footnote n="1">¹A famous illustration of the use of *shadufs* in ancient Egypt was drawn in the Tomb of Neferhotep (Tomb TT49 in Thebes) [12], [18]. Neferhotep lived during the reign of Tutankhamun, Ay and Horemheb during Dynasty XVIII.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 3. The Palermo Stone. Source of the figure: [12], p 135, Fig 2.13.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. The Palermo Stone. Section of the sixth register (sixth row). The level of the Nile is recorded in the lower part of the compartment. The indicated height of the Nile associated with this considered year is 5 cubits 1 palm 1 finger, that is approximately 2.72 m. More details about this particular compartment are well studied in [17], p 44.</figcaption></figure>

their mothers, appear between the rows. The compartments associated with each king indicate his actions during each year of his reign, and in each lower sub-compartment is recorded

the height reached by the Nile flood that year. This structure is displayed in Fig. 3.

In ancient Egypt, the metrological system of Nile flood measurements was cubits, palms and fingers [17] such that

• 1 cubit is approximately equal to 52.5 cm;

• 1 cubit is divided into 7 palms;

• 1 palm is made up of 4 fingers.

A famous historical tool that illustrates this metrological system is the Royal Cubit of Maya [20] illustrated in Fig. 6. In the greatly valuable book [17], Annette Imhausen provides several explanations and examples on how to read this hieroglyphic metrological system. Based on an example studied in [17], we provide in Fig. 4 an illustration of the hieroglyph reading of one year flood level recorded in one section of the Palermo Stone.

In the greatly valuable studies [12], [19], Barbara Bell and Rushdi Said converted into meters and reported all the recovered measurements dating from ancient Egypt. In her paper [19], Barbara Bell focused on the measurements recovered from Dynasty I to Dynasty V. Then, in his book [12], Rushdi Said extended these studies and notably summarized the main recovered measurements dating from ancient Egypt.

### C. Predictions from nilometer data

Thanks to the fairly precision of the dates of its floods and recedings, and thanks to its regular annual contribution of silt throughout its flooding, the Nile was a beneficially predictable river in terms of annual yields. Therefore, in Egyptian agriculture, predictions about the harvest used to be made according to the level of inundation of the Nile using Nilometers data [11], [12], [14]–[17].

In ancient literature, among the historical accounts of these events are those of Diodorus Siculus, an ancient Greek historian of the 1st century BC, known for writing the monumental universal history *Bibliotheca Historica*. In the English translation [16] of the writings of Diodorus Siculus, one finds accounts concerning these events of Nilometer predictions during ancient Egypt (see Diodorus, 1.36.11 - 1.36.12):

Citation from the English translation [16] of Diodorus’s writings 1.36.11 and 1.36.12

"And because of the anxiety occasioned by the rise of the river the kings (of Ancient Egypt) have constructed a Nilometer at Memphis, where those who are charged with the administration of it accurately measure the rise and despatch messages to the cities, and inform them exactly how many cubits or fingers the river has risen and when it has commenced to fall. In this manner the entire nation, when it has learned that the river has ceased rising and begun to fall, is relieved of its anxiety, while at the same time all immediately know in advance how large the next harvest will be, since the Egyptians have kept an accurate record of their observations of this kind over a long period of terms."
D. Nilometers in ancient Egypt and sustainability governance

<figure><img src="image_6.png" /><figcaption>Fig. 5. Model of a Granary with Scribes discovered in the Tomb of the royal chief steward Meketre (TT 280, MMA 1101), Dynasty XII, early reign of Amenemhat I, Middle Kingdom. Source of the figure: The Metropolitan Museum of Art [21].</figcaption></figure>

Thanks to the benefits provided by the Nile floods and the Egyptian agricultural practices, ancient Egypt was considered as the granary of the Mediterranean Basin, which allowed this nation to maintain a long-lasting economy. The economy of ancient Egypt has been described as a “grain economy”, characterized by the collection of grain and its subsequent redistribution [22], [23]. Regarding the collection of grain, a portion of the crops was provided to the administration after the harvest for storage through annual contributions/taxes <fnref n="2" />. The redistribution of crops covered several activities, including the provisions for the government, the administration, the priests, the Temple workers, the workers engaged in state projects, gifts offered for festivals, and a sustainable supply for the population during difficult times, for example during repeated low Nile floods [23]. Indeed, accounts have been recovered in ancient tombs that testimony scenes of grain redistribution during periods of famine, as Barbara Bell mentioned in [24] by referring to [25]. For instance

From the
tomb of
Bebi of
El Kab [24]

“... I was a man who ammassed grain... who was vigilant during the winter. Whenever a famine came, during various/numerous years, I gave grain to my city during each famine...”

From the
tomb of
Horherkhoutef
(Edfu) [24]

“I gave grain to the entire country. I saved my town from famine...”

While the flooding of the Nile was much more predictable and calm than other rivers, the historical measurements illustrate that the volume of the flood could vary from year to year, and some times considerably. If the flooding was insufficient or too high, it could result in poor harvest which could lead to a risk of future famine, economic instabilities and unfair tax pressures [14]. A sound and fair logistics system for harvest collection was therefore necessary for sustainability governance in ancient Egypt, whether for tax assessment, harvest storage (as studied in [26]) and all upstream transport logistics. An illustration of harvest storage is provided in Fig. 5.

As mentioned in [22], [23], tax assessment involved two steps: An initial tax estimation before the harvest and a final tax assessment once the harvest was complete. The practice of estimating taxable capacity before harvest was already noted in the Middle Kingdom and continued into the New Kingdom, as studied in [22], [23]. This estimation of taxable capacity was calculated by estimating the agricultural yields, taking into account several indicators such as the Nile flood level using Nilometer data and plot size. Nilometer predictions played therefore a major role for initial tax estimation and allowed to limit tax pressure, as mentioned Danielle Bonneau in [11]:

English "Nilometers, instruments of observation as

translation well as forecasting, essentially based on empir-ical data, allowed the governances of ancient

from [11] Egypt to estimate, in the least unfair way possible, the limit of tax pressure.”

To summarize, thanks to Nilometer predictions, the adminis-
trations of ancient Egypt were able to adapt their governance
in order to ensure the long-term provision of food for the
population and the state and to reduce the risks of famine.
To this aim, based on the harvest predictions, the levy of
taxes (that is to say the portions of harvests provided to the
administration for storage and redistribution) was calculated
and adapted in advance in the most fair way possible.

IV. THE INTEREST OF STUDYING NILOMETERS IN THE FIELD OF STATISTICAL LEARNING FOR SUSTAINABILITY GOVERNANCE

This section discusses why Nilometers can be a source of
interest for statistical learning in sustainability governance and
which future research could interest both the fields of statistical
learning and sustainability governance.

A. A source of interest for the field of statistical learning

Although several historical sources attest to the use of
Nilometers as instruments for predicting annual crop yields
and associated tax levels, ancient documents on which these
estimates were analytically calculated have unfortunately not
yet been recovered (or if they have, are not yet being

<footnote n="2">²As mentioned in [22], there is no equivalent to the modern word “tax” in the ancient Egypt language, but several words were used to designate different types of collection of revenues for the state, such as for instance “dues”, “tributes”, “harvest-yield” or “gifts”. However, the modern word “tax” is often used to refer to these types of state revenue collection. More details are explained in [22].</footnote>
<figure><img src="image_7.png" /><figcaption>Fig. 6. Royal Cubit of Maya [20]. Maya was *Overseer of the treasuries* during the reign of Tutankhamun, Ay and Horemheb, Dynasty XVIII.</figcaption></figure>

highlighted). Hence, an interesting and important research would be to recovering the mathematical models considered in ancient Egypt for Nilometer predictions. A such discovery would indeed be highly interesting for the field and history of statistical decision theory.

In other words, can we recover the mathematical calculations established using Nilometer data? Referring to Section II, can we recover the kind of decision rule $\delta$ (see equation (1)) considered to predict the annual crop yields based on the Nile floods? In order to obtain accurate predictions, such decision rules should have been established based on past Nilometer data and associated crop yields (as it is for instance accounted in Diodorus' citation "... all immediately know in advance how large the next harvest will be, since the Egyptians have kept an accurate record of their observations of this kind over a long period of terms"). Can we recover documents illustrating how these data were structured? For instance, regarding the Nilometer data engraved in the Palermo Stone, it seems that it was already common to well structure the data (rows and sub-compartments). Can we recover how were stored Nilometer data for establishing the predictive decision rule? Was it different or quite similar to the commonly structured data in statistical learning as for instance in Table I?

Addressing these questions would be highly valuable for the field of statistical learning, providing examples that could be studied whether in education or books as ancient mathematical predictive models or that could be seen as a source of inspiration for future models.

## B. A source of interest face to climate change

Climatology (or climate science) is defined as the scientific study of climate through the analysis of recorded data over a long period. Since climate change impacts both environmental, social and economic dimensions, the climate change of our era will have fundamental causes and consequences in terms of sustainability governance. In order to improve sustainability governance face to climate change, a relevant and important topic is to study similar past phenomena [4], [14], [27]. For instance, as mentioned in [4], bridging the gap between climatology and archaeology could bring several benefits in climate change research. And according to [14], [27], the Nile offers a unique opportunity to study climate change since instrumental data have been recorded for many centuries.

Although we have previously seen that Nilometer predictions played a major role for sustainability governance regarding annual harvest, initial tax estimation and allowed to limit tax pressure, ancient Egypt nevertheless had to overcome certain difficult and unstable periods, whether famines or economic instabilities, particularly during certain climatic

instabilities and critical Nile floods, which could be short-term or longer-term. Indeed, recent geoarchaeological studies illustrate climatic changes during certain periods of ancient Egypt [28], [29], the periods of which in some cases famines and economic instabilities took place during these periods. It was for example the case at the end of the Old Kingdom [14], [29].

As mentioned in [30], Egypt offers an example where it can reasonably be argued that periods (from several decades to several centuries) of stable climatic conditions, alternating with periods of extreme Nile flooding (higher/lower) and periods of more stable (non-extreme) conditions, can either promote or hinder the effectiveness of sustainability management. In [14], the authors studied how instabilities related to Nile floods (extremely high or extremely low) contributed to an increased risk of political instability.

An interesting study that could complement these studies would be to recover how predictions from Nilometer data were used during these difficult times. Although such study appears difficult due to some information not yet retrieved, this could offer the field of statistical learning a historical example of the use of predictive models when extreme events occur, particularly in the fields of climatology and sustainability governance. How some governances managed and succeeded to overcome such environmental/climatic instabilities using these predictive instruments and why was it more difficult for other governances?

## C. A source of interest for sustainability development

As we mentioned in the Introduction, AI models have been more and more referred as a hopeful tool for sustainability development [1]–[3], but also raise concerns regarding trust, ethics, and the role of humans in these models [1]. Among the commonly reported positives impacts, AI models can for instance support sustainable cities, provision of food, health of ecosystems and low-carbon energy systems. However, regarding the commonly reported negative impacts, the growth of AI models could also lead to economic inequalities within nations and also consume a lot of energy in their use.

Regarding these questions and according to the previous sections, ancient Nilometers can therefore provide a relevant source of reflection and inspiration. Indeed, Nilometers aimed to support provision of food and fair tax estimations, without consuming energy in their use thanks to their sustainable engineering. These instruments are also a source of inspiration thanks to their longevity, having played a major role in sustainability governance for millennia, facing significant economic instabilities but remaining very important throughout ancient Egypt since the early measurements of Nile floods.
V. CONCLUSION

The emergence of AI models in sustainability governance is nowadays revealing several promising, positive, questioning and potentially negative impacts. In order to consider a complementary perspective on these questions, a relevant approach is to study how older similar models have behaved in relation to these different aspects, with the aim of being able to draw inspiration from them or improve the negative aspects for the development of future models.

Regarding the field of statistical learning theory, closely related to machine learning algorithms, the considered models are quite recent, the origin of machine learning algorithms emerging in the past century with the advances of computing, and the foundations of statistical learning models raising up in the early 1800s. However, we have seen that the paradigm of prediction based on observed data existed long before the emergence of statistical learning like for instance Nilometers in ancient Egypt.

Prediction based on Nilometer data in ancient Egypt seem to have played a major role in sustainability governance, aiming to support provision of food and fair tax estimations based on the annual Nile floods. These data-based predictive instruments can be a source of inspiration for modern models thanks to their longevity, having played a major role in sustainability governance for millennia, facing significant economic instabilities but remaining very important throughout ancient Egypt since the early measurements of Nile floods.

Several interesting questions remain to be addressed regarding the use of Nilometers. Future research will be devoted to address the questions discussed in Subsection IV-A that could be highly relevant for the field of statistical learning theory, providing ancient examples that could be studied whether in education or books as historical mathematical predictive models or that could be seen as a source of inspiration for future ones.

ACKNOWLEDGMENT

The author would like to thank all the researchers both in archaeology and history of mathematics, for their valuable works helping to recover important sources of information.

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