and made his name a protection and honour to those who could claim any relationship with him. Archimedes was a lover of glory; not of that sordid ambition which inspires mediocrity, but of solid glory, which is due to a man who has enlarged the limits of science. He desired, when he was dying, that a sphere inscribed in a cylinder might be engraved on his tomb, to perpetuate the memory of his most brilliant discovery; the Sicilians, however, having their minds turned upon different objects than Geometry, forgot the man who was their chief honour in the eyes of posterity. Two hundred years after his death, Cicero being then quæstor in Sicily, gave, to use his own words, Archimedes a second time to light: unable to learn from the Sicilians the place of his interment, he sought for it by the symbol before mentioned, and six verses in Greek inscribed upon its base. After much fruitless research, it was at length discovered in a field near Syracuse over-grown with thorns; he showed it to the Sicilians, who blushed for their ignorance and ingratitude. Not more than fifty years had elapsed since the death of Archimedes, when Apollonius arose, who, if not equal to his illustrious predecessor, certainly ranks in the second place among the ancients, and who gave a great impulse to the mathematical sciences. He was born at Perga, in Pamphylia, whence he is called Apollonius Pergæus, to distinguish him from others of the same name. His contemporaries styled him the Great Geometrician, b 2 and posterity has confirmed this honourable title without detracting from the merit of Archimedes, to whom it assigns the first place. Apollonius composed a great number of books, which were considered by the ancients as affording the most perfect examples of the higher geometry of that time; most of these are now lost, or exist only in fragments; we have, however, nearly the whole of his conic sections, which are alone sufficient to establish his fame, and to merit the title before-mentioned; this treatise consisted originally of eight books; the first four of which have been transmitted to us in the language in which they were written; and the following three had been preserved only in an Arabic translation made about the year 1250, and translated into Latin about the middle of the seventeenth century by Borelli; but to the great regret of all geometers, the eighth is entirely lost. A magnificent edition was published by Dr. Halley in folio, at Oxford, in 1710, together with the Lemmas of Pappus, and the Commentaries of Eutacius. The other writings of Apollonius, mentioned by Pappus, are, 1. The Section of a Ratio, or Proportional 2. The Section of a Space, in two books. 4. The Tangencies, in two books. 6. The Plane Lair, in two books. Were I writing a minute history of mathematics, I might give an account of the geome tricians, who flourished from the time of Archimedes to the destruction of the Alexandrian School; but as this introduction is intended only as a brief historical sketch of those ancient mathematicians, who successively improved and made discoveries in the sciences, the reader must not expect to find an enlarged history of an obscure individual, or a full relation of a trifling improvement. It may not be improper, however, to name Conon and Dositheus, both very learned men, and both friends of Archimedes, Gemmius, a mathematician of Rhodes, who wrote a work entitled "Enarrationes Geometrica," &c. After these we may reckon Theodosius, who wrote a treatise on spherics, in which he examines the properties which circles formed by cutting a sphere in all directions have with respect to each other. From the time of this eminent man, we move on for three or four hundred years without meeting with one person who contributed anything to the advancement of the sciences. Theon, however, appeared about 380 years after Christ; and by his skill and perseverance in mathematics and philosophy, he obtained the honourable dignity of being appointed president of the famous Alexandrian School, where, by his erudition and conduct, he gained the greatest respect and reputation. His principal works, which have escaped the ravages of time, are his Scholia, or Notes on Euclid's Elements, and his Commentary on the First Eleven Books of Ptolemy's Almagest. They were published in Greek in the years 1633 and 1638. The Scholia were published by Commandine in one of his Latin editions of Euclid. His Commentaries, however, on the Almagest have not yet been translated, except the first book. One of his most celebrated pupils was his own daughter Hypatia, a very learned and beautiful lady, born at Alexandria about the end of the fourth century. Her father, perceiving her extraordinary genius, had her not only educated in all the ordinary accomplishments of her sex, but instructed in the most abstruse sciences. She made such great progress in philosophy, geometry, astronomy, and the mathematics in general, that she passed for the most learned person of her time. She published Commentaries on Apollonius's Conics, on Diophantus's Arithmetic, and other works. Whilst very young she was chosen to succeed her father in the same school, and to deliver instructions out of that chair, where Am monius, Hierocles, and many other very learned men abounded, both at Alexandria and many other parts of the Roman empire. The pupils of this lovely and surprising female were not less eminent than they were numerous. Amongst whom was the much esteemed Synesius, afterward bishop of Ptolemais. But it was not Synesius only, and the disciples of the Alexandrian School, who admired Hypatia for her virtue and learning: never was woman more caressed by the public, and yet never had woman a inore unspotted character. She was held as an oracle for her wisdom, for which she was consulted by the magistrates on all important cases. In short, when Nicephorus intended to pass the highest compliment on the princess Eudocia, he thought he could not do it better than by calling her another Hypatia. Whilst Hypatia thus reigned the most brilliant ornament of her sex in the annals of history, she was greatly admired by Orestes, the governor of that city, who, on account of her wisdom, often consulted her. This, together with an aversion which Cyril had against Orestes, proved the cause of her ruin. About 500 monks assembling, attacked the governor one day, and would have killed him had he not been rescued by the townsmen; and the respect which Orestes had for Hypatia, causing her to be traduced amongst the Christian multitude, they dragged her from her chair, tore her in pieces, and burnt her limbs. This shocking catastrophe was perpetrated in the Lent of the year 416. For a more particular account of this illustrious victim of fanaticism, see Bossut's History of the Mathematics, English edition, 8vo. 1803. At length we come to Pappus, a consummate mathematician, who flourished towards the end of the fourth century, in the reign of Theodosius the Great many of his works are lost, or lie in the hitherto unexplored recesses of public libraries; Suidas mentioned many of them, as also Vossius de Scientiis Mathematicis: amongst which, his Mathematical Collections, consisting of eight books, have transmitted his name with distinguished lustre to posterity. In them the author has as |