ethica, in 2 books; the other, Sermones, also in 2 books. The whole collection is valuable, both on account of the contents in themselves and also of the numerous passages rescued from destruction only by being inserted therein.

2. John of Stobi cultivated the habit of reading with a pen in his hand. The selections which we have, were arranged, it is said, for the use of his son. Each chapter of the Ecloga and of the Sermones, has its title, under which the extracts are placed, the sources whence they are drawn being noted in the margin. More than five hundred authors are quoted, whose works have mostly perished.-Schöll, vii. 133.

3. The best edition of the Ecloga is Heeren's, Gr. & Lat. Gött. 1792-1801. 4 vols. 8. with dissertations and notes.-Of the Discourses, Gaisford's J. Stobæi Florilegium. Oxf. 1822. 4 vols. 8. 2d ed. 1823-25. with the Lat. vers. of H. Grotius, prolegomena and notes.-of both, Fr. Fabrus (Favre, books. of Lyons), Gr. & Lat. Genev. 1609. fol. — The poetical extracts were collected and edited by H. Grotius. Par. 1623. 4. with a translation in Latin verse. Cf. Schall, vn. 159.

VII.-Mathematicians and Geographers.

§ 202 u. The very name of Mathematics (uaduarα, μaðquariz) is an evidence that their scientific form originated among the Greeks, although the Egyptians and various eastern nations, in earlier times, possessed arithmetical, geometrical, and particularly astronomical knowledge. Arithmetic was in a very incomplete state in Greece before the time of Pythagoras. He was the first who considerably cultivated it; but it was left especially to Euclid to treat the subject scientifically and unite with it the study of geometry. The elements of geometry the Greeks seem to have derived from the Phonicians; although the knowledge which Thales acquired in Egypt is not to be overlooked. The science was afterwards considered as a special means of improving the intellect, and an essential preparatory study for every philosopher. (Cf. § 175.) Hence its great estimation and high cultivation among the Greeks. There are many indications of the use and encouragement which the practical mathematics found among them, especially in connection with mechanical sciences, as Statics, Hydrostatics, and Hydraulics. That the Greeks applied mathematics to architecture, and with the most happy success, uniting the rigid principles of science with the rules of taste, we have sufficient proof in the descriptions of their temples, palaces, porticos, and other edifices, and in the still remaining monuments of that art. Astronomy was introduced by Thales from Egypt. Pythagoras established several principles of this science. Other philosophers exhibited them in a written

form.

§ 203. It is obvious, from what has been said, that mathematical studies in Greece can be traced back only to the two primary schools of philosophy, the Ionian founded by Thales, and the Italic by Pythagoras (§ 168).

From the time of Pythagoras, mathematics, as has been suggested, formed an essential part of philosophy. In the Academy they were specially cultivated; this may be inferred from the inscription (cf. § 175) placed by Plato himself over the door of his school. To the philosophers of this sect the science is much indebted. But in the want of historical evidence, it is impossible to give a definite account of the state of mathematical knowledge during the time preceding Alexander. The names of several mathematicians and astronomers are recorded. The most important are Archytas of Tarentum, inventor of various machines which astonished his contemporaries; Meton of Athens, author of the celebrated lunar cycle (cf. P. V. § 194); and Autoly. cus of Pitane, the most ancient mathematician whose works are preserved.

The works of Autolycus were first published by C. Rauchfuss (Dasypodius). Strasb. 1572. 4. In Lat. trans. by I. Auria, Rome, 1587. 2 vols. 4.-A fragment of a treatise by Archytas, on mathematical science, is found in Porphyry; it was published by J. Gramm. Copenh. 1707. 4.— Cf. Plutarch, Sympos. vii.. and Life of Marcellus.

§ 204. After the time of Alexander, mathematical studies became more

prominent than before. Mathematics were no longer merely a part of philosophy in general, but held the place of a science by themselves. They were cultivated in all the schools, which flourished in this period. The mathematical school of Alexandria was rendered illustrious by the reputation of Euclid, who had a numerous class of disciples, and among them Ptolemy I., the king of Egypt. One of the most distinguished names in this period and indeed in all antiquity, is that of Archimedes of Syracuse, celebrated not only for his successful research into abstract principles, but also for his curious and wonderful mechanical applications and inventions. A third memorable name adorns this period, Apollonius of Perga, whose work on Conic Sections formed an epoch in the history of mathematics. Euclid, Archimedes, and Apollonius, with Diophantus, who lived in the third and fourth century after Christ, may justly be regarded as the great founders of mathematical science.

Other names belong to the period between Alexander and the capture of Corinth; as Heron of Alexandria, author of several treatises on branches of mechanics; Athenæus and Biton, who wrote on military engines and missils; and Philon of Byzantium, who wrote on the same subjects, and to whom is ascribed a work on the seven wonders of the world. Astronomy was cultivated with success in this period, and, according to some, an important influence was exerted by the intercourse with the Babylonians in the expedition of Alexander. Aristarchus of Samos, Eratosthenes of Cyrene, and Hipparchus of Nicæa, are the principal authors of whom we have remains.

Marcos, Astronomie solaire d'Hipparque. Par. 1828. 8.-Wallis, Aristarchus. Oxf. 1688. 8. In the next period, i. e. between the fall of Corinth and the time of Constantine, we find no eminent authors in the pure mathematics. Several writers on astronomical subjects are mentioned; Claudius Ptolemy, in the age of the Antonines, was celebrated above all others. His system of astronomy, as is well known, was much in vogue, and exerted a great influence. Several authors on music, of whom fragments are still extant, are referred to this period; some of them were among the mathematicians of the age; their remains are found in the collection of Meibomius (cited § 208t. 1).-Cf. Scholl, livre vi. ch. xliv.

§ 205. Between the time of Constantine and the overthrow of Constantinople, the list of Greek mathematicians is much larger, but contains few names of great eminence. Diophantus, a contemporary of the emperor Julian, and already mentioned as one of the four ancient fathers of mathematics, is the most important. Pappus and Theon of Alexandria, at the close of the fourth century, may be mentioned next. Hypatia, a daughter of Theon, inherited her father's love of mathematical science; she became a public teacher, and wrote several works which perished in the destruction of the Alexandrian library. Proclus the philosopher wrote on mathematics and astronomy. Leon of Constantinople, in the latter half of the ninth century, is spoken of by the Byzantine historians with much admiration. He was solicited by the Arabian Caliph, Al-mamoun, to remove to Bagdad; the emperor Theophilus, refusing to permit this, opened a public place for Leon to give instruction, and bestowed many honors and privileges upon him. He has left nothing by which we can judge of his merits. We will add only the name of Anthemius of Tralles, in the sixth century, employed by Justinian to construct the church of St. Sophia, of which, however, he only laid the foundation, not living to complete the work. There remains a curious fragment of his work Пegi nagadóžwv μnzavnuárov.—Cf. Schöll, livre v1. ch. xci.

The fragment of Anthemius was published in the Mem. de l'Acad. Inser. et Belles Lettres, vol. XLII. by Dupuy and separately, Par. 1774. 4.--Respecting the celebrated Hypatia, see Menage, Hist. Mulier. Philosoph.-Desvignoles, Dissert. in Bibl. German, vol. 111.-Abbé Goujet, Lett. in Contin. des Memoires de Litt. by Desmolets, vol. v. vI.—Socrates, Hist. Eccles. vii. 15. §206. On the subject of Geography, the knowledge of the Greeks was very limited and imperfect; yet they had writers on the subject, of much value in illustrating the condition of ancient countries.- The earliest work extant is the Periplus of Hanno. Hecataus of Miletus, in his Пɛquinous rus, described the countries known at the time he wrote, in the reign of Darius, about 500 B. C. The Periplus of Scylar has been commonly referred to nearly the same period. The Anabasis of Xenophon may properly be men

It was not

tioned among the geographical works anterior to the time of Alexander, being of great value in relation to upper Asia. Pytheas, of Massilia, a voyager and geographer, probably belonging to the same period, before Alexander, was the author of two works, a description of the ocean and a Periplus. The little now known of them is derived from Strabo and Pliny. · until the period between Alexander and the Roman supremacy, that geography was elevated to the rank of a science. The honor of effecting this is ascribed to Eratosthenes, a very eminent mathematician and scholar, who flourished at Alexandria, B. C. about 230.-Cf. Schöll, livre 1. ch. xviii.; livre Iv. ch. xlv.

§ 207. After the supremacy of Rome, greater advances were made in geographical knowledge. The first distinguished geographer of this period is Strabo, born about 60 B. C., whose work styled Tewyoagiza is a thesaurus comprising nearly the whole history of geography from Homer to Augustus, with all then known upon the subject. The geographical poem of Dionysius of Charax belongs to the age of Augustus. We have a fragment of a work on Parthia, by Isidorus of Charax; published in the reign of Caligula. There are also some geographical pieces under the name of Arrian, who flourished in the reign of Hadrian and the Antonines. But a more important work is that of Pausanias belonging to the same age, and entitled, Itinerary of Greece. The most celebrated of all the ancient writers on geography was Claudius Ptolemy, already mentioned as a mathematician and astronomer about the middle of the second century after Christ. His system of geography remained the only manual in vogue for fourteen centuries. -After Ptolemy, the history of Greek letters presents no author of much importance in this department of study. Before the time of Constantine, Agatharcides of Cnidus, in the latter half of the 2d century, is said by Photius to have written several geographical works; and some extracts are preserved by Photius. We have also a fragment of Dionysius of Byzantium in the second century, and a sort of geographical epitome by a certain Agathemerus, probably of the third century. Of the Byzantine geographers, or those subsequent to Constantine, we may mention as the principal, Marcianus of Heraclea in Pontus, Stephanus of Byzantium, and Cosmas the Egyptian monk.— Cf. Schöll, vol. v. p. 275; vii. p. 33.

208. There are some Greek writers on Tactics, who may be mentioned in this place. The most eminent is Onosander, or Onesander, who lived probably about the middle of the 1st century. He left a work on the military art, in a style remarkably pure for the age; it was a source whence all the later writers on the subject drew materials. Polyænus, a native of Macedonia, a rhetorician or advocate of the 2d century, should probably be mentioned as next in rank, although his work is rather a historical collection of stratagems than a treatise on tactics. Apollodorus, an architect in the time of Trajan, left a work entitled Пologztiza, on military engines. The emperor Adrian is said to have composed a military treatise called Exit devμa, a fragment of which is still extant. Arrian and Elian also left works on the subject of Tactics. The emperor Mauritius, of the 6th century, wrote a treatise on the military art. There are also some treatises written at a later period, which it is not important to specify. Cf. Schöll, vol. v. p. 261. vii. 67.

$208 t. We will now introduce some general references, and then speak of a few distinguished individuals, naming first the mathematicians and after them the geographers.

1. On the history of Mathematics among the Greeks, see references P. I. § 24.-L. Lüders, Pythagoras und Hypatia, oder die Mathematik der Alten. Lpz.1809. 8.-Delambre on the Arithmetic of the Greeks in Peyrard's Archimedes, cited § 210. 5. — The principal mathematical Collections are, that of Thevenot, Vet. Mathemat. Opera. Par. 1693. fol. and that of Wallis, in 3d vol. of his Opera Math. Oxf. 1699. fol.—The following collections of writers on subjects connected with mathematics may be cited.-Astronomical, by Aldus. Ven. 1429. fol.-By Petavins, Uranologion &c. Par. 1630. Amst. 1703. fol. - Musical, by Meursius. Lugd. Bat. 1616. 4. - By Meibomius, Antiq. Musicæ auctores, Gr. & Lat. Amst. 1652. 2 vols. 4.-On Tactics, by Meursius, Gr. & Lat. Lug. Bat. 1613. 4. P. Scriverius, Scriptores rei militaris. Vesa). 1670. 8.-A. H. Baumgartner, Samml. aller Kriegsschriftsteller der Griech. übersetzt &c. Mannh.1779. 2 vols. 4. 2. On the history of Geography among the Greeks, Gosselin, Geographie des Grecs. Par.1790. 3 vols. 4. Blair, cited P. I. § 27.-We may also refer to Malte Brun, and to Mannert and Ukert, cited § 7. 7 (b).—The first collection of Minor Greek Geographers was that of Haschel. Augsb.

1600. 8.-The second, Gronovius. Leyd. 1627. 4.-The third, more complete, Hudson. Oxf.16981712. 4 vols. 8.-Much preparation for a new edition was made by Bredow, before 1812. On his death his apparatus passed into the hands of Spohn and Friedemann, from whom is expected an edition containing all the Greek Geographical remains, excepting those of the four authors sometimes denominated Major, viz. Strabo, Pausanias, Ptolemy, and Stephen of Byzantium.-G. Bernhardy, Geographi Græci Minores, Gr. & Lat. Lpz. 1828. 8. not finished; but very good.

$209. Euclid lived at Alexandria B. C. about 300, in the time of the Egyptian king Ptolemy Soter. His native place is not known. He was a teacher of mathematics, particularly of geometry, in which branch he was the most distinguished scholar among the Greeks.

lu. His Elements (Eroixeiα), in 15 books, were drawn up with great ability, and in a very perspicuous manner. There are two Greek commentaries upon this work, by Proclus and Theon. The latter flourished at Alexandria, in the 4th century (§ 205), and it is only according to his revision of the work that we now possess the Elements of Euclid. The 14th and 15th books are ascribed, and with great probability, to Hypsicles, who lived about the middle of the 2d century. Besides the Elements, we have also several other mathematical pieces ascribed to Euclid.

2. The principal works allowed to be genuine are the Data (Asdouéva), containing geometrical theorems, and the Phenomena (Daivóueva), relating to astronomy.-Schöll, 111. 352.-Fuhrmann, Kl. Handb. p. 339.

3. There have been five editions of the WORKS of Euclid.-Princeps, by S. Grynæus. Bas.1533. fol.-Bas. 1559. fol.-C. Dasypodius (Rauchfuss), Gr. & Lat. Strasb. 1571.-D. Gregory, Gr. & Lat. Oxf. 1703. fol.-Best of all, Peyrard, Gr. Lat. & Gall. Par. 1814. 3 vols. 4.- Of the ELEMENTS, A. Caïano, Gr. & Lat. Rom. 1545. 2 vols. 8.--Ch. Melden. Leyd. 1673. 12.-Th. Haselden, (with the Data). Lond. 1732. 8.-Best, Camerer, Gr. & Lat. Berl. 1824. 8. (1st vol. containing 6 books of the Elements, with Excurs. and Plates.) 2d vol. continued by C. F. Hauber.1726.J. C. Neide. Hal. 1825. 8. good, containing first 6 books, with 10th and 12th. — E. F. August. Berl. 1826-30, 2 vols. 8. critical text.

4. Translations.-There have been many editions of the Elements in Latin; among the best, Birmana. Lpz.1769. 8.-S. Horsley. (12 bks). Oxf.1802. 8.- English.-R. Simpson (bk.1-6, 11,12). Glasg. 1756. 4. and often reprinted.-J. Williamson (whole 15). Lond. 1781-88. 2 vols. 4.- German.-I. F. Lorenz. Hal. 1818. 8. — French.-Peyrard, above cited.

$210. Archimedes was born at Syracuse B. C. about 287, and was put to death by a soldier during the storming and capture of that city by the Roman general Marcellus, B. C. 212. He was celebrated especially for his skill in mechanics; but his inventive genius enriched almost every branch of mathematical science.

1. The sepulchre of Archimedes was near one of the gates of Syracuse, but was forgotten and almost overgrown with briars in the time of Cicero. It was discovered by the exertions of the latter, while Quæstor in Sicily, marked by a small pillar bearing an Iambic inscription and the figures of a cylinder and sphere.-Melot, Vie d'Archimede, and Fraguier, Du tombeau d'Archimede, in the Mem. Acad. Inscr. 11. 321. xiv. 128.

2u. He acquired his greatest celebrity by discovering the relation between the Cylinder and Sphere, and by contriving several military engines, by the aid of which the Syracusans defended themselves for three years against the Romans. We have several works from him; Περὶ τῆς Σφαίρας καὶ Κυλίν Soov, On the Sphere and Cylinder; Kúzhov uétoŋois, The Measuring of the Cirele; Περί των Οχουμένων, Of floating bodies ; Ψαμμίτης Arenarius, and others, In general it may be remarked, however, that we possess the works of Ar chimedes only according to the recensions of Isidorus and his pupil Eutocius in the 6th century.

3. Polybius, Livy, and Plutarch, speak of the engines invented by Archimedes to harass the Romans, but say nothing of his destroying their fleet by means of reflecting-mirrors, or burningglasses, contrived for setting fire to the vessels. Lucian is the first author who mentions the burning of the fleet, but he does not tell the means. Tzetzes and the writers of the Bas-Empire, state that it was by the aid of mirrors. The story has been treated as a mere fable, although the possibility of the thing has been proved by Buffon. - Schöll, 111.360. v11. 57.-Cf. Foreign Rev. No. 1. p. 305.-Edinb. Rev. vol. xv111.-Lond. Quart. Rev. 111. 89, 108.-Gibbon, Rom. Emp. 1v. p. 74. ed. N.Yk. 1822. For an account of the magnificent vessel constructed under the care of Archimedes, for the king of Syracuse, see Schöll, v11. p. 446. cf. P. I. § 167. 2.

4. There have been four editions of the WORKs of Archimedes. - Princeps, by T. Gechauff (printer Hereng), Gr. & Lat. Bas. 1544. fol. Rivault (printer Morel), Gr. & Lat. Par. 1615. fol. repr. 1646. ed. Richard.-Borelli. Messina, 1572. fol. repr. Palerm. 1685. fol. Best entirely, Abr. Robertson (begun by Torelli), Gr. & Lat. Oxf. 1792. fol. with the commentary of Eutocius. → Of the Dimensio circuli (with the Arenarius), Wallis. Oxf.1676. 8.—Arenarius, with Engl, transl, by G. Anderson. Lond. 1784. 8.

German,-Sturm (of the whole Works), Nürmb, 1670. fol. — Hauber, the

Sphere and Cylinder. Tub. 1798. 8.-Krüger, the Arenarius. Quedl. 1820. 8. rard, of whole Works. Par. 1807. 4. 1808. 2 vols. 8.

-French, Peg

$211. Apollonius, surnamed Pergaus from his birth-place Perga in Pamphylia, lived at Alexandria about B. C. 250, under Ptolemy Euergetes. He studied mathematics under those who had been pupils of Euclid.

1u. As a writer he is known by his work on Conic Sections, Kwvizα Zτoizełα, in 8 books. Only the first 4 books, however, are in the Greek; the 3 next are in a Latin translation from an Arabian version, and the 8th exists only as restored by Halley from hints found in Pappus.

2. The 4th, 6th and 7th books of the Conic Sections were translated from the Arabian about the middle of the 18th century, by J. A. Borelli. — The other works of Apollonius were ɛgi 'Eлaqov, De Tactionibus, or Contacts of lines and circles, and 'Eminedo Tono, Planes, which have come to us in a very mutilated state; Пegi Nevoer, De Inclinationibus, of which scarcely anything 、remains; Пɛgi zwo̟tov 'Anoтouis, De Sectione Spatii, of which we have nothing; and Iegi óyoυ'Aлотоus, De Sectione cationis, which is preserved in Arabic.

3. The only edition of the Conics is that of E. Halley, (begun by Gregory,) Gr. & Lat. Oxf. 1710. fol. - Attempts have been made to restore some of the other treatises.-De Tactionibus ; by Camerer. Goth. 1795. 8.-By Haumann. Bresl. 1817. 8.-J. Lawson, the two books of A. concerning Tangencies, &c. Lond. 1795. 4.-On Planes, by R. Simpson. Glasg. 1749. 4. — On Inclinations, by S. Horsley, Gr. & Lat. Oxf. 1770. 4. By R. Barrow. Lond. 1799. 4. De Sections Spati; by E. Halley. Oxf. 1706. 8. with a Latin translation, from the Arabic, of the treatise De Sect. rationis. - By A. Richter, Des Apollonius zwei Bücher von Verhältniss-schnitt (from the Latin of Halley). Elb. 1836. 8.

$212. Pappus, an Alexandrine philosopher and mathematician, flourished in the 4th century. His principal work, known to us, is entitled Maquatızai ovraywyai, Mathematical Collections, in 8 books.

1. This work is chiefly interesting on account of the extracts it contains from mathematical writings, which are lost. Other works are ascribed to him; as, a treatise on military engines, a commentary on Aristarchus of Samos, a work on geography, &c.-Scholl, v11. 49.—Am. Quart. Rev. No. xxl.

2. Only fragments of the Greek text have yet been published. A fragment of the 2d book was published by J. Wallis, in his ed. of Aristarchus of Samos. Oxf. 1688. 8.-The second part of the 5th book, by Eisenmann. Par. 1824. fol. - The preface to the 7th book, by Halley. Öxf. 1706. 8. (with a treatise of Apollonius, as cited (211. 3.) -Some lemmas from the 7th book, in Meibomius, Dialog. de Proportionibus. Hafn. 1655. fol. A Latin version of 6 books (3-8), by Fr. Commandini, an Italian mathematician of the 16th century, printed, Pesaro, 1583. fol. and (ed. Manolessius) Bolog. 1660. fol.-A fragment of the 4th book, not in this version, is given by Bredow, Epistolæ Parisienses. Lpz. 1812. 8.

$213. Diophantus or Diophantes, of Alexandria, lived probably in the 4th century, under Julian. He composed an Arithmetic, › Agıðuntizỳ, in 13 books, of which 6 are now extant. A work styled Περὶ πολυγώνων ἀριθμῶν is also ascribed to him.

1. The Arithmetic of Diophantus is not only important as contributing to the history of Mathematics, by making known the state of the science in the 4th century, but it is also interesting to the mathematician himself, as it furnishes luminous methods for resolving various problems. It presents also the first traces of that branch of the science which was called Algebra, in honor of the Arabian Geber, to whom its invention is ascribed.-Schöll, v11. p. 43.

2. A Latin version of all his remains was published by Xylander (Holzmann). Bas. 1575. fol. The first edition of the text was by C. G. Bachet (de Meziriac), Gr. & Lat. Par. 1621. fol. — A German translation of the treatise Isoi no. úyıð. (von den Polygonal-zahlen) by Poselger. Lpz. 1810. 8.-Of the Arithmetic, by Schultz. Berl. 1822. 8. (containing also Poselger's.)

$214. Hanno, the first name we mention among the geographers, probably lived B. C. about 500. He was a Carthaginian general.

1u. He is supposed to have written in the Punic language the Voyage, which, either during his life or shortly after, was translated into Greek, under the title Iegiloos. What we possess is considered by some as only an abstract of a greater work.

2. The full title is "Αννωνος Καρχηδονίων βασιλέως περίπλοος τῶν ὑπὲρ τὰς Ηρακλέους στήλας Λιβυκῶν τῆς γῆς μερῶν ὧν καὶ ἀνέθηκεν ἐν τῷ τοῦ Κρόνου τεμέ vei dyhovvta Túde. Hanno is represented as sent with a fleet of 60 vessels and 30,000 colonists to explore the western coast of Africa, and as having continued