astronomers of antiquity, was born at Pelusium, in Egypt, about A.D. 70, and died A.D. 147. Though he is not known to have left any writings on Geometry, his great works still extant on Astronomy and Geography, for which he is justly celebrated, supply evidence of their author's having been deeply skilled in the applications of that science. His work on Geography consists of seven books, and his great work on Astronomy of thirteen, entitled "ý μeyáλn ovvtağıs." In the early part of the 9th century it was translated into Arabic under the title of Almagest, a word formed from the Arabic article, and a Greek superlative. Both the original Greek and the Arabic version are still extant. Ptolemy adopted that system of the universe which placed the earth in the centre, and from him it acquired the name of the Ptolemaic system. We may pass over the period which intervened between Ptolemy and Pappus, as no mathematicians of eminence appeared in the school of Alexandria, which at that period was the chief place where philosophy and the sciences were still cultivated. Pappus lived in the time of the Emperor Theodosius, who reigned between 379 and 395 A.D. He was the author of a work entitled "Mathematical Collections," which consists, as its title implies, of Problems and Theorems collected from the works of different mathematicians, with commentaries and historical notices. This work and the Commentary of Proclus are the chief repositories of information respecting the ancient Geometry, and especially the Geometrical Analysis. The work of Pappus originally consisted of eight books, the whole of which are extant in the original Greek except the first book, and the first half of the second book. A translation of the last six books into Latin was made by Commandine, who also wrote a Commentary on the work. These, after his death, were published by the Duke of Urbino, at Pisaurum, in 1588. The first two books of Pappus were not then known to be extant; however, there were found in a MS. of Pappus, in the Savilian Library at Oxford, the last twelve Propositions of the second book. They were translated into Latin by Dr Wallis, and printed at the end of his Aristarchus Samius, in 1688. This fragment being on Arithmetic, it was conjectured that the first two books were on the same subject. The next five books are on Geometry, and the last is chiefly on Mechanics. The following is a very brief account of the contents of the remaining six books of Pappus. The third book discusses four general problems. 1. The solution of the Delian problem, or duplication of the cube by means of the Conic Sections: besides three other solutions, one by Eratosthenes, another by Hero, and a third by himself. There is also a fourth by Nicomedes, by means of the conchoid. 2. A problem respecting the Medietates, a name given to three lines when they were in arithmetical, geometrical, or harmonical proportion. 3. To draw two straight lines from two points in one side of a triangle to a point within it so that they may be greater than the other two sides of the triangle. 4. To inscribe the five regular solids in spheres. The fourth book consists of Theorems. Prop. I. contains an extension of Euclid, 1. 47. Prop. x. is one of the tangencies of Apollonius, to which the three preceding Props. are preliminary. Prop. XIII. On the property of the Arbelon. Prop. xix. On the Spiral of Conon; also a solution of the Delian problem by means of the conchoid, and the trisection of an arc of a circle, with the properties of the quadratrix, and some problems. The fifth book commences with a preface, in which Pappus remarks the instinct of bees whereby they construct their cells on geometrical principles, employing the figure whose base is a regular hexagon, which supplies, with the smallest labour, the greatest possible accommodation. The object of the book is to prove that, of these plane figures which are equilateral and equiangular, and have equal perimeters, the greatest area is contained by the figure with the greatest number of sides; and that of all plane figures of equal perimeters, the circle is the greatest. The subject of isoperimetrical figures is treated in 57 Propositions. It is proved that of plane figures with equal perimeters, the greatest is that which is equilateral and equiangular. This principle is extended to solids. The regular solids are then compared, and it is proved that of those with equal surfaces, the greatest is that with the greatest number of faces. These are introductory to the proposition, that of all solids with equal surfaces, the greatest is the sphere. At the end of the book, it is shewn that there can be only five regular solids, or that only equilateral triangles, squares, and pentagons, can form the boundaries of regular solid bodies. The sixth book is employed chiefly in explaining and connecting some propositions of Theodosius and others in treatises on the sphere, &c. The object of the book is stated in a short preface with reference to the three selected propositions: namely, Prop. 6, Book III. of Theodosius on the Sphere; Prop. 6 of Euclid's Phenomena, and Prop. 4 of Theodosius on Days and Nights. Eight different treatises are quoted or alluded to by Pappus in this book, and it may be considered as peculiarly worthy of notice, that in Props. 31, 32, 33, 34, which are preliminary to some on the sphere, there are stated, as examples, some distinctions of magnitudes which may either be increased or diminished without limit, or may be decreased while there is a limit to the decrease, and conversely. The seventh book is employed on the ancient geometrical Analysis. The preface contains an exposition of the method employed by the ancients in the discovery both of the solution of problems, and the demonstration of theorems. After that follows a particular description of the object and contents of some of the most important treatises of the analytical Geometry of the ancients, the whole of which are said to have consisted of thirtythree books. The seventh book itself consists of Lemmas or subsidiary propositions assumed or employed in the treatises described in the preface. The whole of these thirty-three books existed in the time of Pappus, but the greater part of them have since perished, or at least, are not known to be in existence, either in the original Greek or in any translation. The Data of Euclid, in Greek, two books, de Rationis Sectione in an Arabic version, and seven books of the Conics of Apollonius, four in Greek and three in Arabic, are all that are preserved. The descriptions however of eleven of these books are so particular and entire, that some eminent mathematicians have attempted the restoration of them. Of these attempts some short notices will be found under the names of their respective authors. The treatises of Apollonius, entitled, De Rationis Sectione, De Spatii Sectione, De Sectione Determinatâ, De Tactionibus, and De Inclinationibus, contained the discussion of general problems of frequent occurrence in geometrical investigations, completely solved, and all the possible cases distinguished; also of each case a separate analysis and synthesis were given with determinations in The all cases which required them. The use of these general problems was, the more immediate solution of any proposed geometrical problems, which could be easily reduced to a particular case of some one of them The other treatises in the list were useful for the same purpose. seventh book contains 238 propositions, some of which exhibit complete examples of the ancient analysis and synthesis. The eighth book gives some account of the science of Mechanics, and exhibits the progress it had made in the age of Pappus: it contains many references to the mechanical inventions of Archimedes. A considerable part of the book is employed in describing what are called the five mechanical powers, and the most obvious combinations of them for raising or drawing large weights. Pappus acknowledges that the substance of the book is chiefly borrowed from the works of Hero the Elder, who lived about fifty years after Archimedes. His long preface contains some statements of the mechanical notions and of the arts of that period, as well as some observations on the utility of Mechanics, and on the connexion of Mechanics with Geometry. The branches of the science are distinguished, and some notices are given of treatises which are lost. Serenus lived about the same time, and is chiefly known for his treatise, in three books, on the Cone and Cylinder. Theon, of the same period, was a native of Smyrna: he was a mathematician of the Platonic school at Alexandria, of which he subsequently became president. He wrote a commentary on Euclid's Elements; the earliest of which we have any notice, and another on the first eleven books of the Almagest of Ptolemy. The commentary was translated into Latin by Commandine, and published with his Latin translation of the Elements of Euclid from the Arabic. Hypatia, the daughter of Theon, became so well skilled in the mathematical Sciences as to be chosen to succeed her father in the school at Alexandria. Her commentaries on the Conic Sections of Apollonius, and on the Arithmetic of Diophantus, are not known to be in existence. Though of blameless life, she was assassinated, A.D. 415, and there are some grounds for the opinion, that Cyril the patriarch of Alexandria, was not quite exempt from blame in that horrid deed. Hero the younger was the instructor of Proclus in the mathematical sciences at Alexandria. He was the author of a work on Mensuration, entitled Geodæsia, and another on Mechanics, both of which were published in Latin, at Venice, in 1572. In the Geodesia, there is given the method of finding the area of a plane triangle in terms of the sides of the triangle. Diocles, his cotemporary, discovered the generation of the curve called the cissoid, which still bears his name, and was applied by him in finding two mean proportionals between two given lines. Another solution of this problem was given by Sporus, who lived about the same time. We now come down to the latter times of the Greek school of science and philosophy at Alexandria, which city seems to have been the chief place of refuge for the Grecian sciences. Proclus was born, A.D. 412, at Byzantium, and died at Athens at the age of 75 years. His parents, Patricius and Marcella, were both of Lycian origin, and are spoken of by Marinus as excelling in virtue. Proclus studied first at Alexandria under the most eminent Platonic philosophers. He fre quented the discourses of Olympiodorus, for the purpose of learning the doctrines of Aristotle; and in mathematical science he gave himself up to Hero, whose constant companion he became. He next studied at Athens, where he was the pupil of the celebrated Syrianus, and at length became the chief of the Platonic school established in that city. Of his numerous writings on the Mathematical sciences, his commentary on the first book of Euclid's Elements of Geometry is still extant in the original Greek. It was translated into English by Thomas Taylor, in 1788, and inscribed "To the Sacred Majesty of Truth." The commentary of Proclus, though tinged with much of the mysticism of the later Platonic school, contains some interesting facts relating to the history of Geometry, and many judicious remarks on the definitions, postulates, axioms, and propositions of the first book of the Elements. From a remark at the end, it appears to have been the intention of Proclus, if he had lived, to write commentaries on the other books of Euclid, in a similar style. There are extant two other mathematical works ascribed to him: one is a small treatise on the sphere, which was published in 1620 by Bainbridge, the professor of Astronomy at Oxford: the other is a compendium of Ptolemy's Almagest, entitled Hypotyposis. The original Greek was published in 1540, and a Latin translation by Valla in the following year. Proclus also wrote on many other subjects; commentaries on several dialogues of Plato, of which some are still extant; lectures on Aristotle, and a commentary on the writings of Homer and Hesiod. Four Hymns, one to the Sun, one to the Muses, and two to Venus, are attributed to him. He also wrote on Providence and Fate, and concerning the existence of Evil; besides numerous other pieces, of which we may mention his eighteen arguments against Christianity. These arguments, except the first, are all preserved in the answer of Philoponus. The Greek was published at Venice in 1535, and a Latin version at Lyons in 1557. Marinus of Naples was a disciple of Proclus, and his successor in the school at Athens. There is a commentary still extant, on Euclid's Data, of which he was the author: his other writings on mathematical subjects, though not numerous, have not survived. His most celebrated work, however, is a life of his master Proclus: it was published in 1700, with a Latin version by Fabricius. Of the mathematicians who lived about the middle of the sixth century, (the latest period of the decline of Grecian science,) Eutocius may be regarded as the most distinguished. He was a native of Ascalon in Palestine, and a disciple of Isidorus, one of the architects who designed and built the church at Constantinople, which is now called the Mosque of St Sophia. The only works of Eutocius which have descended to modern times, are two commentaries; one on the Sphere and Cylinder of Archimedes, and the other on the first four books of the Conics of Apollonius Pergæus. The latter was published with the Oxford edition of that author by Dr Halley, in 1710; and the former with the works of Archimedes at Oxford, in 1792. The commentary on the Sphere and Cylinder contains ten various methods of solving the celebrated Delian problem, which are of little importance in the present state of mathematical science. Besides elucidations of difficult passages of the two works, these commentaries contain many useful observations on the historical progress of the mathematical sciences. We must not pass over in silence one writer who lived at the end of the fifth century, and the early part of the sixth, and who was almost the latest author of any eminence that wrote in the Latin language. Boëthius was the most distinguished of the Romans for his scientific writings; which, however, consisted chiefly of translations and commentaries. He was a senator and consul in the reigns of Odoacer and Theodoric, and was put to death by order of Theodoric, A.D. 526. He was educated at Athens; and his writings were numerous on almost every branch of literature and science. He was the author of a treatise on Arithmetic, and another on Geometry: of the latter there is an ancient MS. copy preserved in the Library of Trinity College, Cambridge. He also translated Euclid's Elements, and some of the writings of Archimedes and Ptolemy. Boëthius, however, is chiefly celebrated for his work entitled "De Consolatione Philosophiæ," which was much read in the middle ages, and has been translated into almost all the European languages. An Anglo-Saxon version was made by King Alfred; and ancient MS. copies exist in several public libraries: it was printed at Oxford, in 1698. The rise of the Mahommedan power in the seventh century, and the rapid and desolating conquests which followed, hastened the extinction of the Grecian sciences. In A. D. 640, the Mahommedans invaded and conquered Egypt. The great Library of Alexandria, which is said to have contained at that time many thousand volumes, the writings of geometers, astronomers, and philosophers, was committed to the flames. As a justification of the act, the Khalif Omar declared, that, "if they agreed with the Koran, they were useless, and if they did not, they ought to be destroyed." The learned men who were congregated at Alexandria for the cultivation of science and philosophy, either fell by the swords of the conquerors, or escaped by flight, and these carried with them some remains of the sciences. In somewhat more than a century after this event, the Arabians became the most zealous patrons and cultivators of the science and philosophy of the Greeks and Hindus. The rapid progress of the Mahommedan power, both in the East and in the West, led to the foundation of a powerful empire. The second Abbaside Khalif Almansur ascended the throne A.D. 753, and shortly after, transferred the seat of his government from Damascus to the newly-founded city of Bagdad. Haroun Alrashid, the grandson of Almansur, before his accession to the Khalifat had overrun the Greek provinces of Asia Minor, and penetrated as far as the Hellespont. The reigns of Alrashid and his successor Almamun displayed at Bagdad the highest degree of luxury and splendour, which are depicted in many scenes of the famous tales of the Arabian Nights' Entertainment. The Arabians became acquainted with the astronomical and arithmetical science of the Hindus before they had any knowledge of the writings of the Greek astronomers and mathematicians. It is related in the preface to the Astronomical Tables of Ebn Aladami, that in the reign of Almansur, in the 156th year of the Hegira, A.D. 773, an Indian astronomer visited the court of the Khalif of Bagdad, bringing with him astronomical tables, which, he affirmed, had been computed by an Indian prince whose name was Phighar. The Khalif, embracing the opportunity thus presented to him, commanded the book to be translated into Arabic, and to be published for a guide to the |