:

connection in the order of time and here, accordingly, the highest praise of the ancient philosophers will ever be considered to rest.*

* On this whole subject we refer our readers to the CAB. CYC. " Introductory Essay on the Study of Natural Philosophy," p. 104. et seq.

PART II

THE PROGRESS OF PHYSICAL AND MATHEMATICAL SCIENCE FROM THE MIDDLE AGES TO THE TIME OF NEWTON.

THE decline and dissolution of the Roman empire forms an epoch not more remarkable in the civil and social, than in the intellectual history of mankind. We have observed how entire and universal was the neglect into which all philosophical pursuits and physical enquiries had fallen, during the later portion of those ages which are assigned to ancient history. Here, then, one grand and well defined period marks a corresponding division in the narrative we are attempting to trace; and, as far as Europe is concerned, a long and unhappy interval of general darkness and ignorance succeeded, into which the light of improvement only began to shoot a few tremulous and uncertain rays about the tenth century. Meanwhile, in the East, the great events of this period were accompanied by at least a partial cultivation of science: this, perhaps, might be classed as following in the train of the ancient philosophy; but, nevertheless, we shall prefer considering it under this division, since it had a much closer connection with the modern revival of science.

We shall find, in this period, the following convenient divisions:- The first will comprise the state of science during the middle ages, and its first revival extending to the end of the fifteenth century. The second will include its progress during the next age, as far as to comprehend some of the first grand advances beyond the science of the ancients made by original modern research the age of Copernicus and Tycho. The third will embrace the great discoveries of Kepler and Galileo. The fourth will be occupied by the delivery of

the Baconian philosophy, and the researches of the increasing phalanx of philosophers, who, as it were, prepared the way for Newton during the earlier and middle part of the seventeenth century.

SECTION I.

THE SCIENCE OF THE MIDDLE AGES, AND ITS FIRST RENOVATION, TO THE END OF THE FIFTEENTH CENTURY.

Science in the East during the Middle Ages.

DURING the period of the extinction of science in the Roman empire, the various branches of natural and mathematical knowledge which had long before been cultivated in the oriental regions, though they were not altogether lost or neglected, yet made no progress, and were pursued chiefly by persons of low attainments, and in connection with the arts of astrology and alchemy. Just, however, at the time of its lowest ebb in Europe, the tide of science made a considerable advance in the East. The Arabians had at all times evinced some disposition to astronomical and mathematical studies; and when their warlike tribes had pushed their conquests over a large part of these regions, and had finally established the Arab empire in the East, the throne was occupied by a succession of princes who patronised and encouraged learning. Their conquests put them in possession of the works of the Greek philosophers, which they held in high estimation, and caused translations of them to be made into Arabic. With these aids, as well as the resources of science which they possessed among themselves, the Arabian philosophers devoted increased attention to the cultivation of the mathematical sciences, and particularly astronomy. The prevalence of this taste dates itself from the time of the Caliphs El-Mansour and

Haroun-el-Reschid, a little before A.D. 800; and was at its height under El-Mamoun, soon after that period. This last sovereign ordered a new measurement of the length of a degree, which we are told was performed in the plains of Mesopotamia; but the result is lost to us, owing to an uncertainty about the kind of measure employed. The great work of Ptolemy was also translated into Arabic by his command, under the title of Almagest, A. D. 827.

The two centuries following this æra were fruitful in astronomers, who particularly devoted themselves to accurate observation: the records of their labours are of great extent. Among these Albategnius (or El Batani) discovered the motion of the point of the sun's apogee. He also corrected Ptolemy's determination of the precession of equinoxes.

Ibn Junis made his observations at Cairo about a. D. 1000. It appears that about this time the Arabian astronomers were aware of the properties of the pendulum, and employed it in their observations as a measure of time.

In trigonometry, the Arabian mathematicians certainly introduced some improvements. The most valuable, perhaps, was the substitution of the sine in their calculations instead of the chord used by the Greeks. Aboul Wefa made tables of tangents and cotangents about A. D. 1000. They do not seem to have used the cosine till about a century later, when Geber, a Mahometan Spaniard, gives a formula into which it enters.

It may indeed excite our surprise that such apparently obvious improvements should not have been introduced earlier; and that the speculative genius of the Greek geometry, even without any reference to the practical use of trigonometrical formulæ for astronomical purposes, should not have pursued the innumerable curious relations they present, as matters of abstract contemplation. That the mathematicians of Greece did not do this is, perhaps, to be traced partly to the fastidious dread they always appear to have entertained of mixing up their

geometry with any thing having the appearance of a relation to numerical computation; and it was with an object of this kind in view that trigonometry first presented itself to them. Partly, again, the sort of investigation was in itself of a new class, and of a kind which, perhaps, would not exactly harmonise with the style and manner adopted in their demonstrations.

Yet the whole science is nothing but a continued and highly elegant application of one simple idea, that of giving names to the several ratios subsisting between the sides and angles of a right-angled triangle. The ratio of one of the rectangular sides to the hypothenuse is called the sine of the angle opposite to it, while that of the other side to the hypothenuse is the cosine of the same angle. Then, again, that of the sine to the cosine, is named the tangent; its inverse, the cotangent. These simple names imposed upon ratios often recurring, added to the idea of applying the triangle in different positions, so that its hypothenuse is always the radius, and its sides coincident with the diameters of a circle, whose arc measures the angle, are the elements out of which the whole superstructure of the science is directly reared; and from such simple principles a machine is constructed, which, in the hands of the skilful analyst, has exceeded in power perhaps every other invention of geometry, and has mainly achieved the triumphs of the mathematician in every department to which his science has been applied.

The etymology of the terms above defined has been a subject of question, especially as bearing upon the origin of the science. Some writers have contended for the derivation of the term "sine" from the Arabic. And though others have dwelt upon the Greek and Latin etymologies of the arc, or bow, its chord, and the sagitta, or arrow (a name sometimes applied to the portion of the diameter intercepted), as giving a sort of fanciful description of the diagram, yet, certainly, no meaning of the word "sinus" will very aptly apply to the case.

The Arabs seem to have evinced a considerable taste