The thirteenth century was brilliant when compared with the ages that had gone before ; it was the twilight of that bright day which has enlightened Europe for upwards of 200 years. Among the mathematicians of this time may be reckoned John of Halifax, called also Sacro-Bosco, who wrote a treatise on the sphere, and Campanus of Navarre, the celebrated translator of Euclid, and the author of a trea tise on the quadrature of the circle ; in which he has sup posed that the approximate ratio found by Archimedes was quite exact ; and proceeding on this, he has resolved some problems relating to the circle : IIis paralogism is excusa ble, in consideration of the time in which he lived. The celebrated Albertus Magnus wrote on geometry in this century.
It is instructive to reflect upon the principles in human nature, by which, after ignorance has debased the mind, knowledge is again renovated. In the dark ages, when the true causes which bring about natural events Were un known or but little understood, the principle in the mind, by which men are led to suppose co-existing events as somehow connected, made them conjecture that the mo tions of the heavenly bodies, the most striking phenomena of nature, were closely connected with the common events of life. In this way, probably, astrology became a disease of the mind in the absence of genuine knowledge ; but in pursuit of this delusion, it was necessary to cultivate astro nomy, and this science again required the immediate aid of geometry. Thus we see, that from the very nature of the human understanding, it has a tendency to emerge from ignorance, and that probably we are indebted for the resto ration of the ancient astronomy and geometry to the vain speculations of judicial astrology.
During the 14th century, England was fertile in mathe maticians. They wrote treatises on arithmetic and geome try, but chiefly on astronomy. Their works, however, have chiefly remained in the public libraries. The most re markable was Richard Wallington, who raised himself from an obscure condition by his merit. The science of geometry claims also the poet Chaucer as one of its culti vators. Even at this time, Britain gave indications of the approach of that brilliant xra of discovery, which will for ever render her illustrious among the nations.
The period now approached, in which geometry was to recover more than its original splendour. Its principal pro moters were then Purbach and John Muller, called also Regiomontanus. They greatly improved trigonometry, and
formed the resolution of travelling together into Italy, to study the Greek tongue ; but Purbach dying, Regiomonta nus went alone, and accomplished his purpose. Thus pre pared, he translated the Almagest of Ptolemy from the ori ginal. He also gave Latin versions of the spherics of Me nelaus, those of Theodosius, and his other astronomical treatises : besides, he corrected, by the Greek text, the an cient version of Archimedes made by Gerrard of Cremona. He translated the Conics of Apollonius, the Cylindrics of Serenus, and others of the ancient mathematicians He commented on certain books of Archimedes, which Euto cius had passed over : he defended Euclid's definition of proportionals against Campanus ; and he refuted a pretend ed quadraturc of the circle by Cardinal Cusa.
Purbach rejected the ancient sexigesimal division of the radius, and instead of it he supposed it to be divided into 600,000. Rcgiomontanus, again, improved on Purbach; and, dividing the radius into 1,000,000 parts, he calculated new tables for every degree and minute of the quadrant, adding, for the first time, the tangents. It was Purbach that invented the geometrical square, and he appears to have been the first that applied the plumb line to mark the divisions on instruments.
Locus Pacciolus, or De Burgo, must be reckoned one of the distinguished cultivators of geometry of this period. Ile revised Campanus's translation of Euclid, but his la bours did not appear until 1509. His work, Summa de Arith metica Geometria, &c. 1494, contains a tolerable treatise on geometry. The progress which had now been made in the Greek tongue, and the invention of printing, contributed greatly to the dissemination of geometrical knowledge. The Greek mathematicians began now to be known in Eu rope ; and Euclid was printed for the first time at Venice in 1482, in a folio volume, by Erhard Ratdolt, one of the first printers of the age : its title was, Praclarissimus liber Elementorum Euclidis Perspicacissimiin artem geometria. in ciPit quam felicissime. And at the end we read, Opus Ele mentc,rum Euclidis Megarensis in geometricam artem ; in id quoque Campani perspicassimi commentationes. Erhardus Ratdolt, 4usrustensis impressor Solertissimus, Venetiis im pressit, anno salutis MCCCCLXXX 1I. Oct. cal Junii. Lec tor -vale. On the back of the title-page, there is a dedica tion to the reigning Doge.