MATHEMATICS. 1. GEOMETRY.
great inheritance of mathematical knowledge which the ancients bequeathed to po sterity could not, on the revival of learning, be immediately taken possession of, nor could even its existence be discovered, but by degrees. Though the study of the Mathematics had never been entirely abandoned, it had been reduced to matters of very simple and easy comprehension, such as were merely subservient to practice. There had been men who could compute the area of a triangle, draw a meridian line, or even construct a sun-dial, in the worst of times; but between such skill, and the capacity to understand or the taste to relish, the demonstrations of Euclid, Apollonius, or Archimedes, there was a great inter val, and many difficulties were to be overcome, for which much time, and much subsidi ary knowledge, were necessary. The repositories of the ancient treasures were to be open ed, and made accessible ; the knowledge of the languages was to be acquired ; the manu scripts were to be decyphered ; and the skill of the grammarian and the critic were to pre cede, in a certain degree, that of the geometrician or the astronomer. The obligations which we have to those who undertook this laborious and irksome task, and who rescued the ancient books from the prisons to which ignorance and barbarism had con demned them, and from the final destruction by which they must soon have been overtaken, are such as we can never sufficiently acknowledge; and, indeed, we shall never know even the names of many of the benefactors to whom our thanks are due. In the midst of the wars, the confusion, and bloodshed, which overwhelmed Europe during the middle ages, the re ligious houses and monasteries afforded to the remains of ancient learning an asylum, which a salutary prejudice forced even the most lawless to respect ; and the authors who have given the best account of the revival of letters, agree that it is in a great measure to those establishments that we owe the safety of the books which have kept alive the scientific and literary attainments of Greece and Rome.
The study of the remains of antiquity gradually produced men of taste and intelligence, who were able to correct the faults of the manuscripts they copied, and to explain the dig, ficulties of the authors they translated. Such were Purbach, Regiomontanus, Comman dine, Maurolycus, and many others. By their means, the writings of Euclid, Archi medes, Apollonius, Ptolemy, and Pappus, became known and accessible to men of science. Arabia contributed its share towards this great renovation, and from the language
of that country was derived the knowledge of many Greek books, of the originals of which, some were not found till long afterwards, and others have never yet been discovered.
In nothing, perhaps, is the inventive and elegant genius of the Greeks better exem plified than in their geometry. The elementary truths of that science were connected by Euclid into one great chain, beginning from the axioms, and extending to the properties of the five regular solids ; the whole digested into such admirable order, and explained with such clearness and precision, that no similar work of superior excellence has appeared, even in the present advanced state of mathematical science.
Archimedes had assailed the more difficult problems of geometry, and by means of the method of Exhaustions, had demonstrated many curious and important theorems, with re gard to the lengths and areas of curves, and the contents of solids. The same great geo meter had given a beginning to physico-mathematical science, by investigating several pro positions, and resolving several problems in Mechanics and Hydrostatics.
Apollonius had treated of the Conic Sections,—the Curves which, after the circle,..are the most simple and important in geometry ; and, by his elaborate and profound re searches, had laid the foundation of discoveries which were to illustrate very distant ages.
Another great invention, the Geometrical Analysis, ascribed very generally to the Pla tonic school, but most successfully cultivated by the geometer just named, is one of the most ingenious and beautiful contrivances in the mathematics. It is a method of discover ing truth by reasoning concerning things unknown, or propositions merely supposed, as if the one were given, or the other were really true. A quantity that is unknown, is only to be found from the relations which it bears to quantities that are known. By reasoning on these relations, we come at last to some one so simple, that the thing sought is thereby de termined. By this analytical process, therefore, the thing required is discovered, and we are at the same time put in possession of an instrument by which new truths may be found out, and which, when skill in using it has been acquirdi by practice, may be applied to an unlimited extent.