There was only one mode of escaping such a danger or embarrassment as this, and that was, first, to arrive by repeated observation at an exact knowledge of the route followed by the stars across the sky, and of the rapidity of their march; secondly, to distinguish them one from another, to know each by its own name, to recognize its physiognomy, character, and habits. The first duty of the astrologer was to prepare such an inventory, and to discover the principle of these movements; then, and then only, would he be in a position to give a satisfactory answer to one asking where any particular star would be at the end of any specified number of days, weeks, or months. Thanks to such information, his client could fix upon some happy conjunction of the heavenly bodies, or at least avoid a moment when their influence would be on the side of disaster. In every undertaking of any importance the most favourable hour could be selected long before by the person chiefly concerned, the hour in which his star would be in the best quarter of the sky and in the most propitious relations with its neighbours.
The phenomena produced in Chaldæa by these studies have been repeated more than once in the history of civilization; they embody one of those surprises to which humanity owes much of its progress. The final object of all this patient research was never reached, because the relations upon which a belief in its feasibility was based were absolutely chimerical, but as a compensation, the accessory and preliminary knowledge, the mere means to a futile end, have been of incalculable value. Thus, in order to give an imposing and apparently solid basis to their astrological doctrines, the Chaldæans invented such a numeration as would permit really intricate computations to be made. By the aid of this system they sketched out all the great theories of astronomy at a very early age. In the course of a few centuries, they carried that science to a point never reached by the Egyptians.[93]
The chief difficulty in the way of a complete explanation of the Chaldæan system of arithmetic lies in the interpretation of the symbols which served it for ciphers, which is all the greater as it would seem that they had several different ways of writing a single number. In some cases the notation varied according to the purpose of the calculation. A mathematician used one system for his own studies, and another for documents which had to be read by the public. The doubts attending the question are gradually being resolved, however, by the combined efforts of Assyriologists and mathematicians. At the beginning of their civilization the Chaldæans did as other peoples have done when they have become dissatisfied with that mere rough opposition of unity to plurality which is enough for savage races, and have attempted to establish the series of numbers and to define their properties. "They also began by counting on their fingers, by fives and tens, or in other words by units of five; later on they adopted a notation by sixes and twelves as an improvement upon the primitive system, in which the chief element, the ten, could be divided neither into three nor four equal parts."[94] Two regular series were thus formed, one in units of six, the other in units of five. Their commonest terms were, of course, those that occur in both series. We know from the Greek writers that the Chaldæans counted time by sosses of sixty, by ners of 600, and by sars of 3,600, years, and these terms were not reserved for time, they were employed for all kinds of quantities. The sosse could be looked at either as five twelves or six tens. So, too, with the ner (600) which represents six hundreds, or a sosse of tens, or ten sosses or fifty twelves. The sar may be analysed in a similar fashion.
A system of numeration was thus established which may be looked at from a double point of view; in the first place from its sexagesimal base, which certainly adapts itself to various requirements with greater ease than any other;[95] in the second from the extreme facility with which not only addition, but all kinds of complex calculations may be made by its use.[96]
With but two symbols, one for the units, the other for the tens, every number could be expressed by attending to a rule of position like that governing our written numeration; at each step to the left, a single sign, the vertical wedge, increased sixty-fold in value; the tens were placed beside it, and a blank in this or that column answered to our zero.
Founded upon a sexagesimal numeration, the metrical system of Babylon and Nineveh was "the most scientific of all those known and practised by the ancients: until the elaboration of the French metrical system, it was the only one whose every part was scientifically co-ordinated, and of which the fundamental conception was the natural development of all measures of superficies, of capacity, or of weight, from one single unit of length, a conception which was adopted as a starting point by the French commission of weights and measures."
The cubit of 525 millimetres was the base of the whole system.[97] We shall not here attempt to explain how the other measures—itinerary, agrarian, of capacity, of weight—were derived from the cubit; to call attention to the traces left in our nomenclature by the duodecimal or sexagesimal system of the Babylonians, even after the complete triumph of the decimal system, is sufficient for our purposes. It is used for instance in the division of the circle into degrees, minutes, and seconds, in the division of the year into months, and of the day into hours and their fractions.
This convenient, exact, and highly developed system of arithmetic and metrology enabled the Chaldæans to make good use of their observations, and to extract from them a connected astronomical doctrine. They began by registering the phenomena. They laid out a map of the heavens and recognized the difference between fixed stars and those movable bodies the Greeks called planets—among the latter they naturally included the sun and the moon, the most conspicuous of them all both in size and motion, whose courses were the first to be studied and described. The apparent march of the sun through the crowded ranks of the celestial army was defined, and its successive stages marked by those twelve constellations which are still called the Signs of the Zodiac. In time even these observations were excelled, and it now appears certain that the Chaldæans recognized the annual displacement of the equinoctial point upon the ecliptic, a discovery that is generally attributed to the Greek astronomers. But, like Hipparchus, they made faults of calculation in consequence of the defects of their instruments.[98]
It was the same with the moon. They succeeded in determining its mean daily movements, and when they had established a period of two hundred and twenty-three lunations, they contrived to foretell its eclipses. Eclipses of the sun presented greater difficulties, and the Chaldæans were content with noting their occurrence. They were acquainted with the solar year of three hundred and sixty-five days and a quarter; they used it in their astronomical calculations; but their religious and civil year was one composed of twelve lunar months, alternately full and short, that is, of twenty-nine and thirty days respectively. The lunar and solar years were brought into agreement by an intercalary cycle of eight years.[99]
The assertion of the philosopher Simplicius has been called in question for very plausible reasons. Simplicius declares, upon the faith of Porphyrius, that Callisthenes sent from Babylon to his uncle Aristotle, a copy of Chaldæan observations dating back as far as 1903 years before the entry of Alexander into Mesopotamia, that is, to more than twenty-two centuries before our era.[100]