# Apollonius sentence example

apollonius

- We can trace obligations to Meleager, Theocritus, Apollonius Rhodius and other Alexandrines, and amongst earlier writers to Homer, Pindar, Aeschylus and others.
- Apollonius, the commander of the Syrian garrison in Jerusalem, and Seron the commander of the army in Syria, came in turn against Judas and his bands and were defeated.
- The same difficulty is found in the case of the IIEpi to-Topias referred to by the scholiast on Apollonius.
- Again, others (Apollonius Rhodius) laid down the course as up the Danube (Ister), from it into the Adriatic by a supposed mouth of that river, and on to Corcyra, where a storm overtook them.
- Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated.Advertisement
- It is given at some length in the fourth Pythian ode of Pindar, and forms the subject of the Argonautica of Apollonius Rhodius.
- But there seems no reason for doubt; the great grammarians of imperial times (Apollonius Dyscolus and Herodian) were acquainted with the work in its present form, although, as was natural considering its popularity, additions and alterations may have been made later.
- In recollection of its former services, the emperor Claudius remitted the heavy tribute which had been imposed on it; but the last remnant of its independence was taken away by Vespasian, who, in answer to a remonstrance from Apollonius of Tyana, taunted the inhabitants with having "forgotten to be free."
- The other was written by a certain Apollonius forty years after the appearance of Montanus, consequently about 196.
- A second took place when Vieta pointed to Apollonius's problem of taction as not yet being mastered, and Adriaan van Roomen gave a solution by the hyperbola.Advertisement
- Vieta, however, did not accept it, as there existed a solution by means of the rule and the compass only, which he published himself in his Apollonius Gallus (1600).
- The best extant specimen is the Argonautica of Apollonius Rhodius; the most characteristic is the Alexandra or Cassandra of Lycophron, the obscurity of which is almost proverbial.
- The founder of the mathematical school was the celebrated Euclid (Eucleides); among its scholars were Archimedes; Apollonius of Perga, author of a treatise on Conic Sections; Eratosthenes, to whom we owe the first measurement of the earth; and Hipparchus, the founder of the epicyclical theory of the heavens, afterwards called the Ptolemaic system, from its most famous expositor, Claudius Ptolemaeus.
- His chief poem of the later period was the Argonautae, closely modelled on the epic of Apollonius Rhodius.
- It is a free imitation and in parts a translation of the work of Apollonius of Rhodes, already familiar to the Romans in the popular version of Varro Atacinus.Advertisement
- Priscian informs us in his preface that he has translated into Latin such precepts of the Greeks Herodian and Apollonius as seemed suitable, and added to them from Latin grammarians.
- The latest name in the above list is that of Polybius, who died about 123 B.C. Apollonius Rhodius, Aratus and Theocritus were subsequently added to the " epic " poets.
- Suidas attributes numerous works to him, amongst others a number of letters to Apollonius of Tyana.
- This problem, also termed the " Apollonian problem," was demonstrated with the aid of conic sections by Apollonius in his book on Contacts or Tangencies; geometrical solutions involving the conic sections were also given by Adrianus Romanus, Vieta, Newton and others.
- After the Conics in eight Books had been written in a first edition, Apollonius brought out a second edition, considerably revised as regards Books i.-ii., at the instance of one Eudemus of Pergamum; the first three books were sent to Eudemus at intervals, as revised, and the later books were dedicated (after Eudemus' death) to King Attalus I.Advertisement
- Halley added in his edition (1710) a restoration of Book viii., in which he was guided by the fact that Pappus gives lemmas "to the seventh and eighth books" under that one heading, as well as by the statement of Apollonius himself that the use of the seventh book was illustrated by the problems solved in the eighth.
- The degree of originality of the Conics can best be judged from Apollonius' own prefaces.
- Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.
- Each of these was divided into two books, and, with the Data, the Porisms and Surface-Loci of Euclid and the Conics of Apollonius were, according to Pappus, included in the body of the ancient analysis.
- Vieta thereupon proposed a simpler construction, and restored the whole treatise of Apollonius in a small work, which he entitled Apollonius Gallus (Paris, 1600).Advertisement
- Other works of Apollonius are referred to by ancient writers, viz.
- Whether Victor the Roman bishop and Apollonius the Roman senator ever really made an appearance as Latin authors is quite uncertain.
- Pappus then enumerates works of Euclid, Apollonius, Aristaeus and Eratosthenes, thirty-three books in all, the substance of which he intends to give, with the lemmas necessary for their elucidation.
- The fact that the author of Apollonius is also the author of the Lives of the Sophists is confirmed by internal evidence.
- The Life of Apollonius was first published by Aldus (1502); a French translation by Blaise de Vigenere appeared in 1596; an English translation of the first two books was published in London (1680) by Charles Blount, with some notes by Lord Herbert of Cherbury (prohibited in England in 1693, it was reprinted on the Continent); a full translation appeared in 1903.Advertisement
- In 1749 was published Apollonii Pergaei locorum planorum libri II., a restoration of Apollonius's lost treatise, founded on the lemmas given in the seventh book of Pappus's Mathematical Collection.
- After his death restorations of Apollonius's treatise De sectione determinata and of Euclid's treatise De porismatibus were printed for private circulation in xxv.
- This proposition, which he called the mystic hexagram, he made the keystone of his theory; from it alone he deduced more than 400 corollaries, embracing, according to his own account, the conics of Apollonius, and other results innumerable.
- The only well-known members of the school were Apollonius of Tyana and Moderatus of Gades.
- For members of the school see Apollonius Of Tyana and Moderatus Of Gades.Advertisement
- To these should be added his version from the Arabic (which language he acquired for the purpose) of the treatise of Apollonius De sectione rationis, with a restoration of his two lost books De sectione spatii, both published at Oxford in 1706; also his fine edition of the Conics of Apollonius, with the treatise by Serenus De sectione cylindri et coni (Oxford, 1710, folio).
- Hypatia, according to Suidas, was the author of commentaries on the Arithmetica of Diophantus of Alexandria, on the Conics of Apollonius of Perga and on the astronomical canon (of Ptolemy).
- In his private chapel he had busts of Orpheus, Abraham, Apollonius of Tyana and Jesus Christ.
- The same fear of imbibing the irrational soul of animals, and thereby reinforcing the lower appetites and instincts of the human being, inspired the vegetarianism of Apollonius of Tyana and of the Jewish Therapeutae, who in their sacred meals were careful to have a table free from blood-containing meats; and the fear of absorbing the animal's psychic qualities equally motived the Jewish and early Christian rule against eating things strangled.
- Conjointly with Giovanni Borelli he wrote a Latin translation of the 5th, 6th and 7th books of the Conics of Apollonius of Perga (1661).Advertisement
- The new rgime obviously laid much more stress on the Oriental character of their state, though Philostratus, in his life of Apollonius of Tyana(who visited the Parthian court), states that Vardanes I.
- Two years later Jerusalem was devastated by his general Apollonius, and a Syrian garrison occupied the citadel (Akra).
- In quick succession he overthrew the Syrian generals Apollonius, Seron and Gorgias, and after the regent Lysias had shared the same fate at his hands he restored the Temple worship (165).
- The examination of the miracles of Apollonius of Tyana, professedly founded on papers of Lord Herbert's, is meant to suggest similar considerations with regard to the miracles of Christ.
- The chief authorities for the life of Attila are Priscus, Jordanes, the Historia Miscella, Apollonius Sidonius and Gregory of Tours.Advertisement
- Wilamowitz-Mollendorff ascribes the nucleus of these Scholia to Theon, who wrote similar scholia on Lycophron and Apollonius Rhodius, and is stated to have written a commentary on Theocritus. ?
- In the 3rd century, Hierocles endeavoured to prove that the doctrines and the life of Apollonius were more valuable than those of Christ, and, in modern times, Voltaire and Charles Blount (1654-1693), the English freethinker, have adopted a similar standpoint.
- Apart from this extravagant eulogy, it is absurd to regard Apollonius merely as a vulgar charlatan and miracle-monger.
- He drew from the best authorities - Apollonius Dyscolus, Herodian, Orion, Theodosius of Alexandria.
- On the authority of the two great commentators Pappus and Proclus, Euclid wrote four books on conics, but the originals are now lost, and all we have is chiefly to be found in the works of Apollonius of Perga.Advertisement
- But the greatest Greek writer on the conic sections was Apollonius of Perga, and it is to his Conic Sections that we are indebted for a review of the early history of this subject.
- The remaining books are strikingly original and are to be regarded as embracing Apollonius's own researches.
- The first book, which is almost entirely concerned with the construction of the three conic sections, contains one of the most brilliant of all the discoveries of Apollonius.
- Prior to his time, a right cone of a definite vertical angle was required for the generation of any particular conic; Apollonius showed that the sections could all be produced from one and the same cone, which may be either right or oblique, by simply varying the inclination of the cutting plane.
- To comprehend more exactly the discovery of Apollonius, imagine an oblique cone on a circular base, of which the line joining the vertex to the centre of the base is the axis.
- Apollonius considered sections of the cone made by planes at any inclination to the plane of the circular base and perpendicular to the triangle containing the axis.
- The points in which the cutting plane intersects the sides of the triangle are the vertices of the curve; and the line joining these points is a diameter which Apollonius named the latus transversum.
- This property is true for all conics, and it served as the basis of most of the constructions and propositions given by Apollonius.
- Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.
- The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.
- We may now summarize the contents of the Conics of Apollonius.
- His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola.
- The Conics of Apollonius was translated into Arabic by Tobit ben Korra in the 9th century, and this edition was followed by Halley in 1710.
- Werner was followed by Franciscus Maurolycus of Messina, who adopted the same method, and added considerably to the discoveries of Apollonius.
- Claude Mydorge (1585-1647), a French geometer and friend of Descartes, published a work De sectionibus conicis in which he greatly simplified the cumbrous proofs of Apollonius, whose method of treatment he followed.
- John Wallis, in addition to translating the Conics of Apollonius, published in 1655 an original work entitled De sectionibus conicis nova methodo expositis, in which he treated the curves by the Cartesian method, and derived their properties from the definition in piano, completely ignoring the connexion between the conic sections and a cone.
- Here for two years he was busily engaged in parochial work, but he found time to write articles on "Apollonius of Tyana," on "Cicero" and on "Miracles" for the Encyclopaedia Metropolitana.
- The genius of Cauchy was promised in his simple solution of the problem of Apollonius, i.e.
- In the same volume are treatises on "Geometric Loci, or Spherical Tangencies," and on the "Rectification of Curves," besides a restoration of "Apollonius's Plane Loci," together with the author's correspondence addressed to Descartes, Pascal, Roberval, Huygens and others.
- There is considerable evidence to suggest that Apollonius was seen as an " exalted patriarch " in some sense.
- To honor and worship Apollonius, they erected shrines and other memorials.
- In the days of its greatest power Rhodes became famous as a centre of pictorial and plastic art; it gave rise to a school of eclectic oratory whose chief representative was Apollonius Molon, the teacher of Cicero; it was the birthplace of the Stoic philosopher Panaetius; the home of the poet Apollonius Rhodius and the historian Posidonius.
- Apollonius of Tyana and the so-called Neopythagoreans drew similar ethical consequences from their eclectic study of Plato.
- The 2nd century is the age of the two great grammarians, Apollonius Dyscolus (the founder of scientific grammar and the creator of the study of Greek syntax) and his son Herodian, the larger part of whose principal work dealt with the subject of Greek accentuation.
- At Florence the earliest editions of Homer (1488) and Isocrates (1493) had been produced by Demetrius Chalcondyles, while Janus Lascaris was the first to edit the Greek anthology, Apollonius Rhodius, and parts of Euripides, Callimachus and Lucian (1494-1496).
- Anderson of Aberdeen, in the supplement to his Apollonius Redivivus (Paris, 1612), but by far the best is by Robert Simson, Opera quaedam reliqua (Glasgow, 1776).
- Of these the most famous is Philostratus "the Athenian," author of the Life of Apollonius Tyana, which he dedicated to Julia Domna, wife of Alexander Severus and mother of Caracalla (see Apollonius Of Tyana).
- His investigations are distinguished by their great generality, by the fertility of his resources, and by such a rigour in his proofs that he has been considered the greatest geometrical genius since the time of Apollonius.