Ancient Iraq (42 page)

Mathematics and Astronomy

Our knowledge of Mesopotamian mathematics
¹⁴
is derived from two categories of texts: lists of numbers arranged in various ways (increasing and decreasing series, multiplication and division tables, etc.) and problems. Surprisingly, the majority of these problems are exercises for advanced students (or even possibly intellectual recreations) and not, as one would expect in a so-called ‘primitive’ or ‘archaic’ society, problems relating to architecture, land-surveying, irrigation and other matters of practical interest. The following examples are particularly demonstrative:

The sentence introducing the first problem shows that it is purely hypothetical. The solution is given, but the way to reach it must have been verbally indicated by the teacher. In the second problem, on the cont~ary, the procedure is fully developed. It will be seen that Babylonian mathematicians were fully conversant with cube roots, and this as early as the seventeenth or eighteenth century
B.C
, which is the date of this tablet. They also knew, of course, of square roots and were able to
calculate the square root of 2 with only a very minute error (1.414213 instead of 1.414214). The calculations involved also point to the two main characteristics of Mesopotamian mathematics: they were based on the sexagesimal system, and while all systems of numeration used in antiquity (including the Roman system) were ‘juxtapositional’, they alone used a place-value notation or ‘positional’ system, that is to say a system where the value of a given numeral varies according to its position in the written number. (This is what we do when we write, for instance, 3,333, the same numeral being worth 3,000, 300, 30 and 3 respectively.) Both the sexagesimal and the positional systems offered definite advantages for calculations, but unfortunately, the decimal system was also used within units of the sexagesimal one, and the figure ‘zero’ was unknown until the Seleucid period. The interpretation of Mesopotamian problems is therefore often fraught with difficulty, even for experts, and we must assume that in many cases the students were verbally supplied with the necessary indications.

Another point to be emphasized is that without using symbols Babylonian mathematicians operated by algebraical rather than by arithmetical methods. The terms of many of their problems show that they could only be solved by a process equivalent to the use of quadratic equations. For instance, a problem like
¹⁷

‘I have added 7 times the side of my square and 11 times its surface: the result is 6,15 (in sexagesimal numeration). Write down 7 and 11’

postulates the equation 11
x
²
+ 7
x
= 6,15.

It would also appear from some tablets that the Babylonians were familiar with functions and that their calculations occasionally involved serial, exponential and logarithmic relations. They thought in abstractions; they liked numbers for their own sake, almost forgetting their practical uses. For this reason, their geometry was much less advanced than their algebra.
¹⁸
They were aware of some fundamental properties of the triangle, the rectangle and the circle, but failed in their attempts to demonstrate them and measured polygonal surfaces by rough
approximation. When tablets are inscribed with geometrical figures these are usually meant to illustrate arithmetical problems. Contrary to the Greeks, the Babylonians were less interested in the properties of lines, surfaces and volumes than in the intricate calculations suggested by their mutual relations.

Mathematics found in astronomy a wide field of application and gave this science a degree of precision unrivalled in antiquity.
¹⁹
The need for studying the movements of celestial bodies arose in Mesopotamia from a double preoccupation: metaphysical and chronological. The belief that what happened in heaven was reflected on earth, and the thought that if planets and constellations were identified with gods, kings and countries, and if their mutual relations could be foreseen, it would be possible to predict the future alleviated, to some extent, the dramatic uncertainty which was at the root of Mesopotamian philosophy. Astrology therefore was the foundation of astronomy, although the system adopted was never rigid and left room for divine and human initiative, predetermination in the form of horoscopes appearing only during the Achaemenian period. On the other hand, the Mesopotamians had to solve the problem of the lunar calendar. As far as we can go back in the past, the cycle of the moon had been taken as a convenient means of measuring time. The year began on the first New Moon following the spring equinox and was divided into twelve months of twenty-nine or thirty days. Each day began at sunset and was divided into twelve ‘double hours’ (
bêru
), themselves divided into sixty ‘double-minutes’ – a system which we still follow and owe to the Babylonians. Unfortunately, the lunar year is shorter than the solar year by approximately eleven days, so that after nine years the difference amounts to one full season. Moreover, the lunar month began in the evening when the New Moon was visible for the first time, but those who have lived in Iraq know that the Oriental sky is not always as clear as Europeans imagine and clouds, dust or sandstorms may render this observation impossible. How, then, were the official astronomers to decide that the month had begun, and how
could they calculate in advance the exact date and time at which any month would begin? What were, in other words, the laws of the lunar cycle and – since the motions of the moon are linked with those of the sun – what were, the laws of the solar cycle?

The amazing results obtained by Mesopotamian astronomers in this field are obviously not due to the perfection of their instruments – they had only the gnomon (a rudimentary sundial), the clepsydra (a clock worked by flow of water) and the polos (an instrument registering the shadow projected by a minute ball suspended over a half-sphere). They are due to constant, accurate observation and to the use of mathematics for the extrapolation of the data obtained. At an early date the ‘roads’ of the sun and planets were determined and divided into twelve ‘stations’, themselves divided into thirty degrees (the origin of our Zodiac). We possess observations of Venus (Ishtar) written down under the First Dynasty of Babylon
²⁰
and detailed catalogues of stars from the eighth and seventh centuries
B.C
. Soon, eclipses of the moon and, later, eclipses of the sun could be predicted with a fair degree of accuracy. For centuries the difficulty created by the difference between the solar year and the lunar year was solved arbitrarily, the king deciding that one or two intercalary months should be added to the year, but in the eighth century
B.C
. astronomers remarked that 235 lunar months made up exactly nineteen solar years, and on their advice King Nabû-nasir, in 747
B.C
., decreed the intercalation of seven extra months in nineteen lunar years. The ‘Nabonassar calendar’ became standardized between 388 and 367
B.C
.
²¹
Meanwhile, a considerable amount of work had been done on the preparation of lunar, solar and stellar ephemerides. The tables of new and full moons and of eclipses drawn by Nabû-rimâni (the ‘Naburianus’ of Strabo) at the beginning of the fourth century are incredibly accurate,
²²
and the greatest of all Babylonian astronomers, Kidinnu (Cidenas), who practised in about 375
B.C
., gave the exact duration of the solar year with an error of only 4 minutes and 32.65 seconds. His error in the
value of the motion of the sun from the node was actually smaller than that made by the modern astronomer Oppolzer in 1887.
²³

Admirable as it was, Mesopotamian astronomy lacked what we would call synthesis. Contrary to the Greek astronomers, who lived at the same time as the latest and best of them, the Babylonian astronomers never tried to assemble the numerous data they collected into coherent cosmic theories, such as the heliocentric system of Aristarchus of Samos or the geocentric system of Hipparchus. The reason for this probably was their total submission to the gods, which made them accept the world as it was and not as it could be imagined. Besides, their mind worked differently. To quote a specialist: ‘The Greeks were philosophers as well as geometers, the Chaldaeans were empiricists and sophisticated calculators.’
²⁴
As we shall see, the same defect – if it is one – can also be found in Mesopotamian medicine.

Medicine

No such precision can, of course, be expected from what is still regarded as an art more than a science: medicine. Yet Mesopotamian medicine is worthy of a special study for three main reasons: it is copiously documented, highly interesting and often misunderstood.
²⁵

The Mesopotamians believed that disease was a punishment inflicted by the gods upon men for their sins. The word ‘sin’ should be taken here in a broad sense including not only crimes and moral offences but also small errors and omissions in the performance of religious duties, or the unintentional breaking of some taboo. The offended gods could strike directly. Thus in the Code of Hammurabi, the Babylonian boundary stones and the political treaties of the ancient Near East they are called upon individually to send all kinds of ‘grievous maladies’ upon whoever would destroy or alter the document, and physicians as well as priests recognized the ‘hand’ of various gods in the
symptoms exhibited by the patients. The gods could also allow demons to take possession of the sick person, each demon attacking by preference one part of the body; or they could let a man or a woman fall the victim of a spell cast by a sorcerer or witch. Illness was therefore essentially an ethical defect, a black mark, a condemnation which rendered man morally unclean as well as physically unhealthy; and a moral ailment calling for a moral cure, treatment was in many cases magical and religious. The
bâru
-priest, or diviner, was asked to find out by all the methods at his disposal the hidden sin responsible for the divine wrath, the demons were exorcised by the
âshipu
-priest using magical rites and incantations, and the gods were appeased through prayers and sacrifices.

If Mesopotamian medicine had consisted of nothing but moral catharsis it would hardly deserve its name. But an extensive study of the texts in our possession has shown it under another, entirely different aspect. There were in ancient Iraq true physicians who believed in the supernatural origin of most diseases, but who also recognized the causative action of natural agents such as dust, dirt, food or drink and even contagion; who sometimes referred their patients to the
bâru
or the
âshipu
, but who always observed the symptoms with extreme attention, grouped them into syndromes or diseases and applied chemical or instrumental treatments. Side by side with the sacerdotal and magico-religious medicine (
âshiputu
), there had always been in Mesopotamia a rational and pragmatic medicine (
asûtu
).
²⁶

The physician (
asû
) was neither priest nor witch-doctor, but a professional man belonging to the upper middle-class of the Assyro-Babylonian society. He had spent years at school learning the basic sciences of his time and further years with a senior colleague, mastering his art. The Code of Hammurabi, where nine laws concerning medicine (or rather, surgery) fix the price of certain operations and pronounce mutilation and even death for professional faults (see above, p. 205), gives the impression that the medical profession was controlled by the state and
throws discredit on the ability of its members. But these laws are examples of judgements in exceptional cases, and there is no known instance of their having ever been applied. In fact, physicians were held in high esteem at all times and probably established their own fees. Consultants of renown were in great demand, and we know that royal courts exchanged doctors. Thus Tushratta, King of Mitanni in the fourteenth century, sent physicians to the pharaoh Amenophis III, and medical experts from Babylon were dispatched to the Hittite monarch Hattusilis III (1275 – 1250
B.C.
).

We possess a considerable number of lists of symptoms and medical prescriptions written by physicians and several letters addressed to or sent by doctors. From numerous tablets or fragments of tablets written between the eighth and the fifth centuries
B.C.
but belonging to the same series Professor Labat has been able to reconstruct a complete ‘treatise’ of medical diagnosis and prognosis.
²⁷
It comprised forty tablets and was divided into five ‘chapters’. The first chapter, intended in fact for exorcists, gave an interpretation of the ominous signs which could be observed when proceeding to the patient's house. Thus:

When the exorcist proceeds to the patient's house… if he sees a black pig, this patient will die; (or) he will be cured after extreme suffering… If he sees a white pig, this patient will be cured; (or) he will be in distress… If he sees a red pig, this patient will [die?] on the 3rd month (or) on the 3rd day…
²⁸

Then came the description of a variety of symptoms grouped together by organs, by syndromes or diseases, and by order of occurrence. A last group of six tablets was devoted to gynaecologi-cal and infantile diseases. Throughout the treatise emphasis was placed on prognosis rather than on diagnosis proper, and treatments were rarely indicated. Similar texts, or collections of texts, dealt only with the diseases of certain organs, others were more particularly concerned with therapeutics. Here are a few examples chosen among the diseases which can readily be identified:

Several texts were devoted to psychiatric diseases, including ‘depression’ which is not as modern as one may think.
³²
While the diagnosis and prognosis of Mesopotamian physicians were a mixture of superstition and accurate observation, their therapeutics owed nothing to magic.
³³
The oldest ‘pharmacopaeia’ known to date is a collection of recipes dating from the Third Dynasty of Ur, which describes the preparation of ointments, lotions and mixtures made from minerals and plants and might have been written two or three hundred years ago. Drugs were administered in every possible way short of injections: mixtures, potions, inhalations, fumigations, instillations, ointments, liniments, poultices, enemas, suppositories. It is often impossible for us to identify some of the simples and salts which entered into their composition, but in many cases ingredients that have only recently fallen into disuse or that are still used in pharmacy can be recognized. In the following recipe, for instance, opium by mouth and emollients in local application are prescribed for urinary retention:

Crush poppy seeds in beer and make the patient drink it. Grind some
myrrh, mix it with oil and blow it into his urethra with a tube of bronze. Give the patient anemone crushed in
alappanu
-beer.
³⁴

And here is the complex, though rational formula of a poultice to be applied in case of ‘stricture of the lungs’:

Take… parts of the kidney of a sheep; ½
qa
of dates, 15
kisal
of firtree turpentine, 15
kisal
of pinetree turpentine, 15
kisal
of laurel, 13
kisal
of opopanax, 10
kisal
of resin of galbanum, 7
kisal
of mustard, 2
kisal
of cantharis… Grind these drugs in a mortar together with fat and dates. Pour the mixture on a gazelle's skin. Fold the skin. Put it on the painful area and leave it in place for three days. During that time the patient shall drink sweet beer. He shall take his food very hot and stay in a warm place. On the fourth day, remove the poultice, etc…
³⁵

In some cases the physician acted instrumentally. In a letter to Ashurbanipal the king's personal physician, Arad-Nanna, expresses his views on the treatment of epistaxis:

As regards the nose-bleeding… the dressings are not properly applied. They have been placed on the side of the nose, so that they interfere with respiration and the blood flow into the mouth. The nose should be plugged up to its end that the air entry be blocked, and the bleeding will cease.
³⁶

Modern physicians would not change a word of this procedure.

Finally, we must quote an amazing text which proves that, contrary to general belief, the Mesopotamians had some notions of hygiene and preventive medicine. Zimri-Lim, King of Mari, who lived
c
. 1780
B.C
. once wrote to his wife Shibtu:

I have heard that the lady Nanname has been taken ill. She has many contacts with the people of the palace. She meets many ladies in her house. Now then, give severe orders that no one should drink in the cup where she drinks, no one should sit on the seat where she sits, no one should sleep in the bed where she sleeps. She should no longer meet many ladies in her house. This disease is contagious (
mushtahhiz
, from the verb
ahâzu
, to catch).
³⁷

Thus Mesopotamian medicine, although still shrouded in superstition, had already some features of a positive science. Transmitted in part to the Greeks, together with Egyptian medicine, it paved the way for the great Hippocratic reform of the fifth century
B.C.
Yet in its two thousand years of existence it made very little progress. The physicians of Mesopotamia, like her astronomers, founded their art upon metaphysical doctrines and thereby closed the door to a fruitful quest for rational explanations. They knew the answers to many of the ‘whens’ and ‘whats’, but they lacked the curiosity to ask themselves ‘how?’ and ‘why?’. They never attempted to build up theories, but modestly – and perhaps wisely – devoted their efforts to the collection of data. It is only fair to say that they often surpassed in their achievements the other learned men of the ancient East.