astronomer Nasseer-ad-deen flourished, known by his tables and other works (about A. D. 1220). Among

the successors of Timour, though at a considerably later period (A. D. 1440), Ulugh Beg, sovereign of Samarcand, established one of the largest observatories recorded in history, and fitted it up with various instruments, especially quadrants of very great radius. With the help of many scientific assistants he published elaborate collections of tables, and a catalogue of the stars. Their accuracy has been admitted to be considerably superior to that of the Greek tables.

The same patronage was extended to science by the successors of Gengis-khan on the throne of China. The thirteenth century may be considered, owing to this cause, and the importation of the astronomy of the western nations, as the most brilliant epoch of Chinese science.

The subject of the quadrature of the circle engaged the attention of the Arabian mathematicians, who carried somewhat further the approximation by which Archimedes and Apollonius had expressed the ratio of the diameter to the circumference. They gave the ratio as that of 1250 to 3927, which is a very near approach. Another value in still simpler terms, and nearly as accurate, was afterwards given by Metius, which is easily remembered, from its containing only the first three odd numbers, each repeated twice in order; viz. 113 to 355.

This subject is, in fact, one which has continued to exercise the speculations of men in a most singular degree in all ages. It seems to stand forth as an exception to the general truth of the indifference manifested towards the abstract researches of geometry. Here we have a problem which, as far as it is connected with any practical application, has ages ago received such a solution as to suffice amply for all purposes of computation in which it can be introduced; and which later mathematicians have found means of expressing, by a decimal continued to any number of places of

figures which the accuracy of our calculation may demand a problem undistinguished in this respect by any peculiarity, but which stands only as one among many others of the same class, and in the same predicament. And yet, without any conceivable practical object in view, it has in all ages, since its first introduction, obtained an unexampled celebrity even among the most ignorant classes; and has called up a host of investigators who have devoted themselves to its solution with an ardour absolutely bordering on insanity; and which in many instances, if it did not commence, actually terminated in mental aberration. Entirely ignorant of the real object to which such an enquiry ought to be directed, they have pursued their pretended solutions by almost any path except that to which the very conditions of the problem restrict it, viz. strictly geometrical demonstration by elementary methods. That to attain this is absolutely impossible, we have every degree of assurance short of actual demonstration. It certainly has not been demonstrated that the circumference and diameter of a circle are incommensurable; but every argument from analogy and probability leads us to believe it. By other methods than those of strict geometry the problem is solved, and the pretended methods are therefore superfluous.

We shall see at a future stage of our enquiry how much good has been done indirectly by those discussions; in that, from the agitation of the question, real mathematicians have been led to examine the various forms and laws of the series in which the approximations to these and other values of the same kind may be carried on. But the fame which the problem has acquired among ignorant empirics has probably arisen from some connection it was supposed to have with we know not what mystical secrets; hence the persevering diligence with which they have been always going on proposing, one after another, every variety of childish conceits, dignified with the name of demonstrations: some measuring the length of the line gone over during one re

volution of a wheel, others emptying the contents of a cylindrical into a cubical vessel; one cutting out the areas and weighing them, another turning the four quadrants of a circle inside out; some finding the quadrature in the stars, others in the mystical number of the beast in the Apocalypse.

This mighty secret, and that of the philosopher's stone, the mysteries of alchemy, and, above all, of judicial astrology, appear to have obtained considerable influence in the ages of which we are treating, especially among the Arabians. Indeed the astronomy of this people, zealously and ingeniously as it was pursued and cultivated, would appear to have been generally associated with astrology, and made subservient to it. The whole theory of its predictions was built upon the configurations of the planets with respect to the sign of the zodiac which was rising at the moment of an individual's birth (and which was called his horoscope). This, of course, depended on a tolerably exact knowledge of the planetary motions; and, to construct the horoscope for any person, required tables of those motions, and a facility of calculating the places of the heavenly bodies at any given epoch. There can be little doubt that the belief in these absurdities produced at least one powerful motive for the cultivation of astronomy, both in the age of which we are speaking, and even long afterwards among the Europeans.

The occult arts had found their way into Europe at an earlier period, and various circumstances conspire to show that they originated in the East. Astrology was certainly unknown in Greece before the expeditions of Alexander; and it appears to have first made its appearance in Rome in the time of Augustus, when an immense influx of Orientals into the capital took place, as we well know from the satirical remarks of Juvenal.

From that time downwards, even to the seventeenth century, the whole of the civilised world in the West, as well as the East, was enslaved by this extraordinary and superstitious delusion; and, even at this day, we

have hardly got rid of the traces of it from some of our almanacs.

Science in Europe in the Middle Ages.

During the middle ages in Europe, the cultivation of science had sunk to its lowest ebb. Not only was there no original energy displayed in enlarging the stores of knowledge, but there did not exist even the desire to retain what had been already known. There were no writers able to add to the stock of information, and few readers even capable of informing themselves of what their predecessors had done. The records of ancient learning and philosophy were a sealed book, from the total ignorance which prevailed of the Greek language; and though there existed a few Latin versions, or rather compendiums, of one or two of the treatises of Aristotle, and perhaps of some other works of science, either derived from the oriental translations, or possibly from the remains of the literature of the Roman empire, yet these were little read, and still less appreciated or understood. The little knowledge which existed was confined exclusively to the clergy and the monks; and if, in any instance, this extended to more than what barely sufficed for the discharge of their ecclesiastical functions, the cases were rare exceptions. The only philosophy professed was an adherence to the dogmas of Aristotle; and, probably, this profession was made by numbers who knew no more of his opinions than a few technical phrases, and little of his works except their titles.

The monastic establishments were, however, the sanctuaries which afforded an asylum to learning, such as it was; and, unquestionably, whatever remains of ancient literature have been preserved to us, we owe to them.

In ages like those of which we are speaking, when nothing but universal barbarism, tumult, and bloodshed prevailed around, the monasteries were generally held sacred by all parties; and amidst the ignorance which extended without, the more powerfully did superstition

guard these sanctuaries from spoliation. In the libraries attached to these institutions, the manuscripts of the ancient writers were at least preserved from the violence of military plunder, and the wanton devastation of hostile incursions. The works of the philosophers, and the precious records of the mathematical and astronomical discoveries of past ages, might here at least have reposed in undisturbed security from external injury. But, unfortunately, they were not safe from destruction of another kind; and there can be no doubt that numbers of these invaluable remains of antiquity perished in supplying materials to the monkish transcribers, who ruthlessly erased the pages of ancient science and literature to insert the legends of the saints. In process of time, probably, they would all have met with this ignoble fate; but before that period could arrive, the dawn of a brighter age had begun to appear: and a few individuals arose among the monastic orders superior to the general cha racter of their brethren, who were at least sufficiently aware of the value of some of the treasures contained in these repositories, to rescue them from such unworthy and barbarous destruction.

The mere act of preservation of these precious remains of antiquity, is one which must entitle those who performed it to the gratitude of posterity, and would alone suffice to give them a claim to the title of enlightened and discriminating men, superior to the general intelligence of their contemporaries. But between a bare apprehension that these volumes possessed some value, and the taste and ability requisite to appreciate justly their contents, there was, doubtless, a considerable interval. And, indeed, where such taste existed, there were still many difficulties in the way of its cultivation; for the work of completely laying open the treasures of ancient science, much time and much subsidiary knowledge were necessary. The repositories of these valuable stores were to be made accessible; the knowledge of the language in which they were written was to be ac quired; the manuscripts were to be decyphered; and