succession of great French mathematicians, for example, G. Monge,
Géométrie descriptive (1800); J. V. Poncelet, Traité des proprietés
projectives des figures (1822); M. Chasles, Aperçu historique sur
l’origine et le développement des méthodes en géométrie (Bruxelles, 1837),
and Traité de géométrie supérieure (Paris, 1852); and many others.
But the works which have been, and are still, of decisive influence on
thought as a store-house of ideas relevant to the foundations of
geometry are K. G. C. von Staudt’s two works, Geometrie der Lage
(Nürnberg, 1847); and Beiträge zur Geometrie der Lage (Nürnberg,
1856, 3rd ed. 1860).
The final period is characterized by the successful production of exact systems of axioms, and by the final solution of problems which have occupied mathematicians for two thousand years. The successful analysis of the ideas involved in serial continuity is due to R. Dedekind, Stetigkeit und irrationale Zahlen (1872), and to G. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leipzig, 1883), and Acta math. vol. 2.
Complete systems of axioms have been stated by M. Pasch, loc. cit.; G. Peano, loc. cit.; M. Pieri, loc. cit.; B. Russell, Principles of Mathematics; O. Veblen, loc. cit.; and by G. Veronese in his treatise, Fondamenti di geometria (Padua, 1891; German transl. by A. Schepp, Grundzüge der Geometrie, Leipzig, 1894). Most of the leading memoirs on special questions involved have been cited in the text; in addition there may be mentioned M. Pieri, “Nuovi principii di geometria projettiva complessa,” Trans. Accad. R. d. Sci. (Turin, 1905); E. H. Moore, “On the Projective Axioms of Geometry,” Trans. Amer. Math. Soc., 1902; O. Veblen and W. H. Bussey, “Finite Projective Geometries,” Trans. Amer. Math. Soc., 1905; A. B. Kempe, “On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points,” Proc. Lond. Math. Soc., 1890; J. Royce, “The Relation of the Principles of Logic to the Foundations of Geometry,” Trans. of Amer. Math. Soc., 1905; A. Schoenflies, “Über die Möglichkeit einer projectiven Geometrie bei transfiniter (nichtarchimedischer) Massbestimmung,” Deutsch. M.-V. Jahresb., 1906.
For general expositions of the bearings of the above investigations, cf. Hon. Bertrand Russell, loc. cit.; L. Couturat, Les Principes des mathématiques (Paris, 1905); H. Poincaré, loc. cit.; Russell and Whitehead, Principia mathematica (Cambridge, Univ. Press). The philosophers whose views on space and geometric truth deserve especial study are Descartes, Leibnitz, Hume, Kant and J. S. Mill. (A. N. W.)
GEOPONICI,[1] or Scriptores rei rusticae, the Greek and Roman
writers on husbandry and agriculture. On the whole the Greeks
paid less attention than the Romans to the scientific study of
these subjects, which in classical times they regarded as a branch
of economics. Thus Xenophon’s Oeconomicus (see also Memorabilia,
ii. 4) contains a eulogy of agriculture and its beneficial
ethical effects, and much information is to be found in the writings
of Aristotle and his pupil Theophrastus. About the same time
as Xenophon, the philosopher Democritus of Abdera wrote a
treatise Περὶ Γεωργἰας, frequently quoted and much used by
the later compilers of Geoponica (agricultural treatises). Greater
attention was given to the subject in the Alexandrian period;
a long list of names is given by Varro and Columella, amongst
them Hiero II. and Attalus III. Philometor. Later, Cassius
Dionysius of Utica translated and abridged the great work of
the Carthaginian Mago, which was still further condensed by
Diophanes of Nicaea in Bithynia for the use of King Deïotarus.
From these and similar works Cassianus Bassus (q.v.) compiled
his Geoponica. Mention may also be made of a little work
Περὶ Γεωργικῶν by Michael Psellus (printed in Boissonade,
Anecdota Graeca, i.).
The Romans, aware of the necessity of maintaining a numerous and thriving order of agriculturists, from very early times endeavoured to instil into their countrymen both a theoretical and a practical knowledge of the subject. The occupation of the farmer was regarded as next in importance to that of the soldier, and distinguished Romans did not disdain to practise it. In furtherance of this object, the great work of Mago was translated into Latin by order of the senate, and the elder Cato wrote his De agri cultura (extant in a very corrupt state), a simple record in homely language of the rules observed by the old Roman landed proprietors rather than a theoretical treatise. He was followed by the two Sasernae (father and son) and Gnaeus Tremellius Scrofa, whose works are lost. The learned Marcus Terentius Varro of Reate, when eighty years of age, composed his Rerum rusticarum, libri tres, dealing with agriculture, the rearing of cattle, and the breeding of fishes. He was the first to systematize what had been written on the subject, and supplemented the labours of others by practical experience gained during his travels. In the Augustan age Julius Hyginus wrote on farming and bee-keeping, Sabinus Tiro on horticulture, and during the early empire Julius Graecinus and Julius Atticus on the culture of vines, and Cornelius Celsus (best known for his De medicina) on farming. The chief work of the kind, however, is that of Lucius Junius Moderatus Columella (q.v.). About the middle of the 2nd century the two Quintilii, natives of Troja, wrote on the subject in Greek. It is remarkable that Columella’s work exercised less influence in Rome and Italy than in southern Gaul and Spain, where agriculture became one of the principal subjects of instruction in the superior educational establishments that were springing up in those countries. One result of this was the preparation of manuals of a popular kind for use in the schools. In the 3rd century Gargilius Martialis of Mauretania compiled a Geoponica in which medical botany and the veterinary art were included. The De re rustica of Palladius (4th century), in fourteen books, which is almost entirely borrowed from Columella, is greatly inferior in style and knowledge of the subject. It is a kind of farmer’s calendar, in which the different rural occupations are arranged in order of the months. The fourteenth book (on forestry) is written in elegiacs (85 distichs). The whole of Palladius and considerable fragments of Martialis are extant.
The best edition of the Scriptores rei rusticae is by J. G. Schneider (1794–1797), and the whole subject is exhaustively treated by A. Magerstedt, Bilder aus der römischen Landwirtschaft (1858–1863); see also Teuffel-Schwabe, Hist. of Roman Literature, 54; C. F. Bähr in Ersch and Gruber’s Allgemeine Encyklopädie.
GEORGE, SAINT (d. 303), the patron saint of England, Aragon
and Portugal. According to the legend given by Metaphrastes
the Byzantine hagiologist, and substantially repeated in the
Roman Acta sanctorum and in the Spanish breviary, he was born
in Cappadocia of noble Christian parents, from whom he received
a careful religious training. Other accounts place his birth at
Lydda, but preserve his Cappadocian parentage. Having embraced
the profession of a soldier, he rapidly rose under Diocletian
to high military rank. In Persian Armenia he organized
and energized the Christian community at Urmi (Urumiah),
and even visited Britain on an imperial expedition. When
Diocletian had begun to manifest a pronounced hostility towards
Christianity, George sought a personal interview with him, in
which he made deliberate profession of his faith, and, earnestly
remonstrating against the persecution which had begun, resigned
his commission. He was immediately laid under arrest, and
after various tortures, finally put to death at Nicomedia (his body
being afterwards taken to Lydda) on the 23rd of April 303. His
festival is observed on that anniversary by the entire Roman
Catholic Church as a semi-duplex, and by the Spanish Catholics
as a duplex of the first class with an octave. The day is also
celebrated as a principal feast in the Orthodox Eastern Church,
where the saint is distinguished by the titles μεγαλόμαρτυρ and
τροπαιοφόρος.
The historical basis of the tradition is particularly unsound, there being two claimants to the name and honour. Eusebius, Hist. eccl. viii. 5, writes: “Immediately on the promulgation of the edict (of Diocletian) a certain man of no mean origin, but highly esteemed for his temporal dignities, as soon as the decree was published against the churches in Nicomedia, stimulated by a divine zeal and excited by an ardent faith, took it as it was openly placed and posted up for public inspection, and tore it to shreds as a most profane and wicked act. This, too, was done when the two Caesars were in the city, the first of whom was the eldest and chief of all and the other held fourth grade of the imperial dignity after him. But this man, as the first that was distinguished there in this manner, after enduring what was likely to follow an act so daring, preserved his mind, calm and serene, until the moment when his spirit fled.” Rivalling this anonymous martyr, who is often supposed to have been St George, is an earlier martyr briefly mentioned in the Chronicon Pascale: “In the year 225 of the Ascension of our Lord a persecution of the Christians took place, and many
- ↑ The latinized form of a non-existent Γεωπονικοί, used for convenience.
