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governor, and the phenomena of miracles are required to from a Latin inscription, printed by Meyer (Anthologia evidence the genuineness of the prophetic mission. For Latina, 278), which has been supposed to refer to him. man, in order to his well-being and the permanence of his He is in all probability the Festus who was proconsul in kind, requires in the first place a clear vision of right and Africa in 366 and following years, and in Achaia in 372. truth, and must, secondly, depend upon some power capable He is the author of the following works: 1. Descriptio of carrying out these discoveries of moral law. If provi- Orbis Terræ, sometimes called Metaphrasis Periegeseos dence has so arranged that the eyelids and the hair of the Dionysii, being derived from the replnymous of that writer; eyebrows shall grow to protect the eye, much more is it 2. Ora Maritima, of which there is extant only a fragneedful for a prophet to arise who shall preach the truth ment describing the Atlantic coast, and the Mediterranean of God's unity, prescribe laws for men, and exhort them to as far as Marseilles; 3. Aratea Phaenomena, and Aratea well-doing by the promise of recompense to come. The Prognostica, which are paraphrases of two works of Aratus. weal of humanity demands the revelation from God, and, These poems, with the exception of the Aratea, are conto certify his office, the prophet must work miracles. Just tained in Wernsdorf's Poëtæ Latini Minores, vol. v. pt. ii. as in ordinary states the soul influences the bodily organs, AVIGLIANO, a town of Italy, in the province of so in exalted conditions it may attain the level of those Basilicata, 11 miles N.N.W. of Potenza. It stands on the high immaterial spirits, whose energy is strong enough to declivity of a hill, and contains a collegiate church, several permeate the whole passive world. This mystical union convents, and a royal college. A peculiar kind of pottery with the hidden universe is a mystery which the ordinary produced here towards the end of the 18th century is still mind cannot understand. Many things then become visi- sought after by collectors. The surrounding country is ble as by a lightning flash in the darkness, and are appre said to produce the finest cattle in the kingdom. A part hended by the vigorous grasp of pure intuition. But of the town was destroyed by a land-slip in 1824. Popumore generally the imagination throws itself on these lation, 15,982. intuitions, and presents them to the lower soul under the AVIGNON, the chief town of the department of semblance of forms and sounds—the angelic beauty which | Vaucluse in France, situated in a beautiful plain, on the seer beholds, and the harmonious speech which a the left bank of the Rhone, not far from the entrance of heavenly voice seems to utter in his ear. Thus Avicenna, the Durance. It is surrounded by its ancient crenellated like his predecessors, tried to harmonize the abstract forms walls, which are in a state of remarkable preservation, of philosophy with the religious faith of his nation. But and, on the outside, by a line of pleasant boulevards his arguments are generally vitiated by the fallacy of as- planted with trees. 'A precipitous rock rises from the suming what they profess to prove. His failure is made river's edge; and from its summit the cathedral of Notre obvious by the attack of Algazali on the tendencies and Dame des Doms, a building of the 12th century, looks results of speculation.
down on the city, but is almost thrown into insignificance Upwards of 100 treatises ar ascribed to Avicenna. Some of by the Palace of the Popes, which rises by its side, and them are tracts of a few pages, others are works extending through several volumes. The best-known amongst them, and that to which Avicenna owed his European reputation, is the Canon of Medicine; an Arabic edition of it appeared at Rome 1593, and a Hebrew version at Naples in 1491. Of the Latin version there were about thirty editions, founded on the original translation by Gerard of Cremona. The 15th century has the honor of composing the great com. mentary on the text of the Canon, grouping around it all that theory bad imagined, and all that practice had observed. Other medical works translated into Latin are the Medicamenta Cordialia, Canticum de Medicina, Tractatus de Syrupo Acetoso. Scarcely any member of the Arabian circle of the sciences, including theology, philology, mathematics, astronomy, physics, and music, has been left untouched by the treatises of Avicenna, many of which probably varied little, except in being commissioned by a different patron and having a different form or extent. He wrote at least one treatise on alchemy, but several others have been falsely attributed to him. His book on animals was translated by Michael Scot. His Logic, Metaphysics, Physics, De Calo, are treatises giving a synoptic view of Aristotelian doctrine. The Logic and Metaphysics have been printed more than once; the latter, e.g., at Venice in 1493, 1495, and 1546. Some of his shorter essays on medicine, logic,
Sketch-Plan of Avignon. &c., take a poetical form (the poem on logic was published by Schmelders in 1836). Two encyclopædic treatises, deal. 1. Palace of the Popes. 2. Former Palace of the Archbishops. 3. Town, ing with philosophy, are often mentioned. The larger, Al- House. 4. Calvet Museum. 5. Convent of the Visitation. 6. Theological Shefa (Sanatio), exists nearly complete in manuscript in Seminary, (St. Charles). 7. Hospital (St. Louis). 8. Cavalry Barracks. the Bodleian Library and elsewhere; part of it on the De
9. Barracks. 10. Penitentiary. i1. Infantry Barracks. 12. St. Joseph's
College. 13. Convent of the Holy Sacrament. 14. Hotel-Dieu and Anima appeared at Pavia (1490) as the Liber Sextus Natu
General Charity. 15. Church of St. Symphorien. 16. Church of the ralium, and the long account of Avicenda's philosophy given Sacred Heart. '17. Prisons. 18. Savings Bank and Load Office. 19 by Sbahrastani seems to be mainly an analysis, and in many Court-House. 20. Lyceum. 21. Lyceum. 22. Suspension Bridge. 23. places a reproduction, of the Al-Shefa. A shorter form of Benezet Bridge. A, Place du Palais. B, Place de l'Hôtel de Ville. C, the work is known as the Al-Nedjat (Liberatio). The Latin
Rue de la République. D, Rue Calade. F, Place du Corps Saint. G, editions of part of these works have been modified by the
Rue des Lices. H, Place Pie. J. Vieux Septier. K, Rue du Saule. L
Rue Carrêterie. M, Porte du Rhône. N, Porte de la Ligne. 0, Porto corrections which the monkisb editors confess that they ap- St. Lazarus. Q, Porte L’Imbert. R, Porte St. Michael. s, Porte St. plied. There is also a Philosophia Orientalis, mentioned by Roche. T, Porte de l'Oulle. Roger Bacon, and now lost, which according to Averroes was pantheistic in tone.
stretches in sombre grandeur along the southern slope. For Avicenda's life, see Ibn Khallikan's Biographical Dic. This building, or congeries of buildings, was commenced by tionary, translated by Slane (1842); Wüstenfeld's Geschichte Benedict XII. in 1336, and continued by successive popes der Arabischen Aerzte und Naturforscher, Göttingen, 1840; for sixty years. It covers an area of rather more than 17 Abul-Pharagius, Historia Dynastiarum. For his medicine, see
acres. The paintings with which it was profusely adorned Sprengel, Histoire de la Médecine ; and for his philosophy, see sbabrastani, Germ. transl. vol. ii
. 213-332; Prantl, Geschichte are in great measure destroyed, and even the grandeur der Logik, ii. 318-361; Stöckl, Phil. d. Mittelalters, ii. 23-58 ; of its dismantled interiors was for a long time broken Munk, Mélanges, 352-366; and Haneberg in the Abhandlungen in upon by the carpentry and plaster-work of French der Philos. Philolog. Class. der Bayerischen Academie, 1867.
barracks. A restoration has, howerer, been for some time (w. w.) in progress; and the building will again be appropriated
for ecclesiastical and civic purposes. The churches of St. AVIENUS, RUFUS FESTUS, a Latin poet, who appears Agricol, St. Didier, and St. Pierre may be mentioned as of to have flourished in the latter half of the 4th century. some importance; also the papal mint, now known as a Ang knowledge we have of the facts of his life is derived music academy; the town-hali, built in 1862; the Calvet museum, rich in Roman remains; the Requien museum of Adaja, about 3000 feet above the sea-level, at the termina natural history; and the Hôtel des Invalides. Of the tion of the Guadarrama Mountains. “On all sides," says church of the Cordeliers, in which Petrarch's Laura was a recent traveller, "the town is surrounded by a tawny buried, only a small part is standing, and the tomb itself desert, over whose arid plains numbers of gray boulders has been entirely destroyed. The city is the seat of an are scattered like flocks of sheep." Its ancient wall is still archbishop, and has tribunals of primary jurisdiction and in good preservation, crowned by a breastwork, with towers commerce, a royal college, a theological seminary, a society of great strength; but a large part of the town lies beyond of arts, the Vaucluse academy, a public library, a theatre, the circuit. Avila is the seat of a bishop suffragan to &c. The chief object of industry is the preparation of Santiago, and has a Gothic cathedral, built by Garcia silk and the manufacture of silk goods; there are also de Estrella in 1107; a number of interesting churches, such manufactures of paper, leather, hats, jewelry, iron-ware, as Santo Tomas, with the beautiful tomb of Prince Juan, &c. Avignon is remarkably subject to violent winds, of San Vincenti, with its remarkable carving, and Nuestra which the most disastrous is the mistral ; and, according Seraf. Madre Santa Teresa, built over the birthplace of the to the proverb, Avenio ventosa, sine vento venenosa, cum patroness of Spain (who here founded the convent of St. vento fastidiosa (windy Avignon, liable to plague when it Joseph); as well as several monasteries and schools, an has not the wind, and plagued with the wind when it has infirmary, and a foundling hospital. It was formerly the it). The town was a place of some importance in the seat of a university, which was founded in 1482, and times of Roman supremacy, and seems to have had some changed into the college of St. Thomas in 1807. The only special connection with the Greek colony at Massilia. It manufacture of any importance is the spinning of the was incorporated with the Burgundian kingdom, and on wool furnished by the native sheep. Population, 6892. its dissolution became a free republic, after the Italian type. AVILA, GIL GONZALEZ D', a Spanish biographer and As late, indeed, as 1790, it retained its consuls, though its antiquary, was born at Avila about the year 1577, and republican constitution was really destroyed by Charles of died there in 1658. He was made historiographer of Castile Anjou. From 1309, when Clement V. took up his abode in 1612, and of the Indies in 1641. Of his numerous in the city, to 1377, when Gregory XI. returned to Rome, works, the most valuable are his Teatro de las Grandezas A vignon was the seat of the papal court, and it continued de Madrid (Madrid, 1623, sqq.), and his Teatro Edlesiastico, from 1378 to 1418 to be the seat of French anti-popes. In descriptive of the metropolitan churches and cathedrals 1348 it was purchased by Pope Clement VI. from Joanna of Castile, with lives of the prelates (Madrid, 1645–53, 4 of Sicily for the sum of 80,000 florins, and it remained in vols. 4to). possession of the popes till the French Revolution. Popu- AVILA Y ZUNIGA, LUIS D', author of a Spanish lation in 1872, 38,196.
history of the wars of Charles V. Nothing is known as AVILA, a province of Spain, one of the modern divis- to the place or date either of his birth or of his death. He ions of the kingdom of Old Castile, situated between was probably of low origin, but married a wealthy heiress long. 4° 14' and 5° 55' W., and lat. 40° 48' and 41° 18' N. of the house of Zuniga, whose ame he added to his own. It is bounded on the N. by Valladolid, E. by Segovia and He rose rapidly in the favor of the Emperor Charles V., Madrid, S. by Toledo and Caceres, and w. by Salamanca. served in the army and as ambassador to Rome, and The area is 2570 square miles; population, 176,769. It was present at the funeral of Charles in 1558. His work naturally divides itself into two sections, differing com- is entitled Comentarios de la Guerra de Alemaña, hecha de pletely in soil, climate, productions, and social economy. Carlos V. en el año de 1546 y 1547, and appears to have The northern portion is generally level ; the soil is of indif- been printed in 1548. It became very popular, and was ferent quality, strong and marly in a few places, but rocky translated into English, French, Dutch, German, Italian, in all the valleys of the Sierra de Avila; and the climate and Latin. As was to be expected from the position of the alternates from severe cold in winter to extreme heat in author, the book gave a rather one-sided account of Charles, summer. The population of this part is agricultural. The and its misrepresentations have been severely criticised. southern division is one mass of rugged granitic sierras, inter- AVILES, SAN NICOLAS DE (the Latin Flavionavia), a spersed, however, with sheltered and well-watered valleys, town of Spain, in the province of Oviedo, about a league abounding with rich vegetation. The winter here, especially from the sea-coast, in lat. 43° 34' N., long. 5° 58' W. It in the elevated region of the Paramera and the waste lands has a considerable trade by means of its port, which affords of Avila, is long and severe, but the climate is not unhealthy. good anchorage for all classes of vessels. There are here The inhabitants are occupied in the rearing of cattle. The some copper works and coal mines, and the stone quarries are principal mountain chains are the Guadarrama, separating extensive and productive. Aviles has two parish churches, this province from Madrid; the Sierras de Avila, a con- a theatre, and a public school. Population, 3297. tinuation of them westward; the Sierra de Gredos, running AVLONA, or Valona (the ancient Avláv), a town and from the south of Piedrahita through Barco, Arenos, and seaport of Albania, in the eyalet of Yanina. It stands on part of Cebreros; and the Paramera, stretching southwards an eminence near the Gulf of Avlona, an inlet of the from the city of Avila into Arenas and Cebreros. The Adriatic, almost surrounded by mountains.
The port, various ridges which ramify from the latter are covered which is protected by the island of Sasseno, the ancient with wood, presenting a striking contrast to the bare peaks Saso, is the best on the Albanian coast. It is visited of the Sierra de Gredos, and the barren levels in which weekly by Austrian steamers, and carries on considerable they rise on the north. The principal rivers are the intercourse with Brindisi, &c. The town is about a mile Alberche and Tietar, belonging to the basin of the Tagus, and a half from the sea, and has rather a pleasant appearand the Tormes, the Corneja, and the Adaja, belonging to ance with its minarets and its palace, surrounded with that of the Douro. The mountains contain silver, copper, gardens and olive-groves. The Christian population, of iron, lead, and coal, but their mineral wealth has been which a considerable proportion are Italians, is largely exaggerated, and the actual production is absolutely nil. engaged in commerce; while the Turks manufacture Quarries of fine marble and jasper exist in the district of woollen stuffs and arms. The material imported into Arenas. The province has declined in wealth and popula- England for lanning, under the name of Valonia, is the tion during the last two centuries, a result due less to the pericarp of an acorn produced in the district. Avlona want of activity on the part of the inhabitants than to the played an important part in the wars between the Normans oppressive manorial and feudal rights and the strict laws and the Byzantine empire. In 1464 it was taken by the of entail and mortmain, which have acted as barriers to Ottomans; and after being in Venetian possession in 1690, improvement. The principal production is the wool of the was restored to them in 1891. In 1851 it suffered severely Merino sheep, which at one time yielded an immense from an earthquake.
Game is plentiful, and the rivers abound in fish, AVOIRDUPOIS, or AVERDUPOIS, the name of a system specially trout. Olives, chestnuts, and grapes are grown, of weights, commonly supposed to be derived from the and the culture of silk-worms is also carried on. There French, avoir du pois, to have weight. The suggested is little trade, and the manufactures are few, consisting derivation from averer, to verify, seems, however, more chiefly of copper utensils, lime, soap, cloth, paper, combs, probable, averdupois being the earlier form of the word. &c. The state of elementary education is comparatively Avoirdupois weight is used for all commodities except the good, and the ratio of crime is proportionately low (Madoż, precious metals, gems, and medicines. The pound avoirDiccionario de España).
dupois, which is equal to 7000 grains troy, or 453-54 AVILA (the ancient Abula), a city of Spain, the capital of grammes, is divided into 16 ounces, and the ounce into 16 the above province, is situated on the right bank of the drams. See WEIGHTS AND MEASURES
AVOLA, a city on the coast of Sicily, in the province port both in general philosophy and in special science; it of Syracuse, with 11,912 inhabitants. It manufactures also has passed into the language of common life, being straw-mats, and has trade in wine, grain, oil, honey, &c.; applied to any assertion of the truth of which the speaker and there are sugar plantations.
happens to have a strong conviction, or which is put forAVON, the name of several rivers in England, Scotland, ward as beyond question. The scientific use of the word is and France. The word is Celtic, appearing in Welsh as most familiar in inathematics, where it is customary to lay afon, in Manx as aon, and in Gaelic as abhuinn (pronounced down, under the name of axioms, a number of propositions avain), and is radically identical with the Sanskrit ap, water, of which no proof is given or considered necessary, though and the Latin aqua and amnis. The root appears more or the reason for such procedure may not be the same in every less disguised in a vast number of river names all over the case, and in the same case may be variously understood by Celtic area in Europe. Thus, besides such forms as Evan, different minds. Thus scientific axioms, mathematical or Aune, Anne, Ive, Auney, Inney, &c., in the British Islands, other, are sometimes held to carry with them an inherent we have Ajf and Aven in Brittany, Avenza and Avens in authority or to be self-evident, wherein it is, strictly speak. Italy, Avia in Portugal, and Avono in Spain; while the ing, implied that they cannot be made the subject of formal terminal syllable of a large proportion of the French rivers, proof; sometimes they are held to admit of proof, but not such as the Sequana, the Matrona, the Garumna, and so within the particular science in which they are advanced on, seems originally to have been the same word. The as principles; while, again, sometimes the name of axiom names Punjab, Doab, &c., show the root in a clearer shape. is given to propositions that admit of proof within the (See Taylor's Words and Places.) Of the principal Eng- science, but so evidently that they may be straightway aslish rivers of this name in its full form three belong to sumed. Axioms that are genuine principles, though the basin of the Severn. The Upper or Shakespearean raised above discussion within the science, are not thereAvon, rising in Northamptonshire, near the battlefield of fore raised above discussion altogether. From the time of Naseby, flows through Warwickshire, Worcester, and Aristotle it has been claimed for general or first philosophy Gloucester, past Rugby, Warwick, Stratford, and Evesham, to deal with the principles of special science, and hence and joins the larger river at Tewkesbury; while thé have arisen the questions concerning the nature and origin Lower Avon has its sources on the borders of Wiltshire, of axioms so much debated among the philosophic schools. and enters the estuary of the Severn at King's Roads, after Besides, the general philosopher himself, having to treat passing Malmesbury, Bath, and Bristol. (See Ireland's of human knowledge and its conditions as his particular Upper Avon; Lewis's Book of English Rivers, 1855.) The subject-matter, is called to determine the principles of cerMiddle or Little Avon has its whole course in Gloucester- titude, which, as there can be none higher, must have in a shire, and reaches the Severn a short distance below the peculiar sense that character of ultimate authority (howtown of Berkeley. Another river of this name rises in ever explicable) that is ascribed to axioms; and by this Wilts, and flows past Salisbury to the British Channel. In name, accordingly, such highest principles of knowledge Scotland one is a tributary of the Clyde, another belongs have long been called. In the case of a word so variously to the basin of the Forth, and a third joins its waters with employed there is, perhaps, no better way of understanding the Annan, while an Aven is a confluent of the Spey: In its proper signification than by considering it first in the France there are two “Avons” in the system of the Loire, historical light-not to say that there hangs about the and two in that of the Seine.
origin and early use of the name an obscurity which it is AVRANCHES (ancient Abrincata, or Ingena), a town of importance to dispel. of France, in the department of Manche. It was an im- The earliest use of the word in a logical sense appears portant military station of the Romans, and has in more in the works of Aristotle, though, as will presently be modern times sustained several sieges, the most noticeable shown, it had probably acquired such a meaning before of which was the result of its opposition to Henry IV. It his time, and only received from him a more exact deterstands on a wooded hill, commanding a fine view of the bay mination. In his theory of demonstration, set forth in the and rock of St. Michel, about three miles distant. At the Posterior Analytics, he gives the name of aíiom to that imfoot of the hill flows the river Sée, which at high tide is mediate principle of syllogistic reasoning which a learner navigable from the sea. The principal trade is in corn, must bring with him (i. 2, 8); again, axioms are said to be cider, and salt; and candles, lace, nails, parchment, leather, the common principles from which all demonstration takes &c., are manufactured, Avranches was formerly a bishop's place-common to all demonstrative sciences, but varying see; and its cathedral, destroyed as insecure in the time in expression according to the subject-matter of each (i. of the first French Revolution, was the finest in Normandy. 10, 4). The principle of all other axioms—the surest of Its site is now occupied by an open place, called after the all principles-—is that called later the principle of Contracelebrated Huet, bishop of Avranches; and one stone re- diction, indemonstrable itself, and thus fitted to be the mains with an inscription marking it out as the spot where ground of all demonstration (Metaph., iii. 2, iv. 3). ArisHenry II. received absolution for the murder of A Becket. totle's followers, and, later on, the commentators, with Saint-Saturnin's church dates from the 13th century, and glosses of their own, repeat his statements. Thus, accordhas a remarkable gateway. The ancient episcopal palace ing to Themistius (ad Post. Anal.), two species of axioms is now used as a museum of antiquities; and an extensive were distinguished by Theophrastus-one species holding public library is kept in the "mairie." A new cathedral of all things absolutely, as the principle (later known by is in course of erection. The agreeable situation and cli- the name) of Excluded Middle, the other of all things of mate of this town make it a favorite residence of English the same kind, as that the remainders of equals are equal. families. Population in 1872, 8137.
These, adds Themistius himself, are, as it were, connate AXHOLM, or AXELHOLM, an island in the
N.W. part and common to all, and hence their name Axiom; " for of Lincolnshire, England, formed by the rivers Trent, Idle, what is put over either all things absolutely or things of and Don. It consists mainly of a plateau of slight elevation, one sort universally, we consider to have precedence with and comprises the parishes of Althorpe, Belton, Epworth, respect to them.” The same view of the origin of the name Haxey, Luddington, Owston, and Crowle; the total area reappears in Boethius's Latin substitutes for it—dignitas being about 47,000 acres. At a very early period it would and maxima (propositio), the latter preserved in the word appear to have been covered with forest; but this having Maxim, which is often used interchangeably with Axiom. been in great measure destroyed, it sank into a compara- In Aristotle, however, there is no suggestion of such a tive swamp. In 1627 King Charles I.
, who was lord of the meaning. As the verb aflowy changes its original meaning island, entered into a contract with Cornelius Vermuyden, of deem worthy into think fit
, think simply, and also claim or a Dutchman, for reclaiming the meres and marshes, and require, it might as well' be maintained that aziwua-rendering, them fit for tillage. This undertaking led to which Aristotle himself employs in its original ethical the introduction of a large number of Flemish workmen, sense of worth, also in the secondary senses of opinion or who settled in the district, and, in spite of the violent dictum (Metaph., iii. 4), and of simple proposition (Topics, measures adopted by the English peasantry to expel them, viii. 1)—was conferred upon the highest principles of retained their ground in sufficient numbers to affect the reasoning and science because the teacher might require physical appearance and the accent of the inhabitants to them to be granted by the learner. In point of fact, later This day. Élaborate volumes have been published on the writers, like Proclus and others quoted by him, did attach island by Peck (1815), Stonehouse, and Read. (See paper, to Axiom this particular meaning, bringing it into relation by E. Peacock, in Anthropological Review, 1870.)
with Postulate (airnua), as defined by Aristotle in the AXIOM, from the Greek agiwua, is a word of great im- | Posterior Analytics, or as understood by Euclid in his Elements. It may here be added that the word was used | familiar one, that they are indemonstrable principles of regularly in the sense of bare proposition by the Stoics construction and demonstration respectively. Proclus him(Diog. Laert., vii. 65, though Simplícius curiously asserts self (412-485 A.D.) practically comes to rest in this distincthe contrary, ad Epict. Ench., c. 58), herein followed in tion, and accordingly extrudes from the list of postulates later times by the Ramist logicians, and also, in effect, by all but the three received in modern times. The list of Bacon.
axioms he reduces to five, striking out as derivative the That Aristotle did not originate the use of the term two that assert inequality (4th and 5th), also the two that axiom in the sense of scientific first principle, is the nat- assert equality between the doubles and halves of the same ural conclusion to be drawn from the reference he makes respectively (6th and 7th). Euclid's postulate regarding to "what are called axioms in mathematics" (Metaph., iv. the equality of right angles and the other assumed in the 3). Sir William Hamilton (Note A, Reid's Works, p: doctrine of parallel lines, now printed as the 11th and 12th 765) would have it that the reference is to mathematical axioms, he holds to be demonstrable: the 10th axiom (reworks of his own now lost, but there is no real ground for garded as an axiom, not a postulate, by some ancient such a supposition. True though it be, as Hamilton urges, authorities, and so cited by Proclus himself)—Two straight that the so-called axioms standing at the head of Euclid's lines cannot enclose a space-he refuses to print with the Elements acquired the name through the influence of the others, as being a special principle of geometry. Thus he Aristotelian philosophy, evidence is not wanting that by restricts the name axiom to such principles of demonstrathe time of Aristotle, a generation or more before Euclid, tion as are common to the science of quantity generally. it was already the habit of geometricians to give definite These, he then declares, are principles immediate and selfexpression to certain fixed principles as the basis of their manifest—untaught anticipations whose truth is darkened science. Aristotle himself is the authority for this asser- rather than cleared by attempts to demonstrate them, tion, when, in his treatise, De Colo, iii. 4, he speaks of the The question as to the axiomatic principles, whether of advantage of having definite principles of demonstration, knowledge in general or of special science, remained where and these as few as possible, such as are postulated by it had thus been left by the ancients till modern times, mathematicians (καθάπερ άξιούσι και οι εν τοις μαθήμασιν), when new advances began to be made in positive scientific who always have their principles limited in kind or num- inquiry and a new philosophy took the place of the periber. The passage is decisive on the point of general math- patetic system, as it had been continued through the ematical usage, and so distinctly suggests the very word Middle Ages. It was characteristic alike of the philoaxiom in the sense of a principle assumed or postulated, sophic and of the various scientific movements begun by that Aristotle's repeated instance of what he himself calls Descartes to be guided by a consideration of mathematical by the name-If equals be taken from equals, the remain- method—that method which had led in ancient times to ders are equal-can hardly be regarded otherwise than as special conclusions of exceptional certainty, and which a citation from recognized mathematical treatises. The showed itself, as soon as it was seriously taken up again, conclusion, if warranted, is of no small interest, in view more fruitful than ever in new results. To establish philoof the famous list of principles set out by Euclid, which sophical and all special truth after the model of mathehas come to be regarded in modern times as the typical matics became the direct object of the new school of specimen of axiomatic foundation for a science.
thought and inquiry, and the first step thither consisted in Euclid, giving systematic form to the elements of geomet- positing principles of immediate certainty whence deducrical science in the generation after the death of Aristotle, tion might proceed. Descartes accordingly devised his propounded, at the beginning of his treatise, under the criterion of perfect clearness and distinctness of thought name of ôpoi, the definitions with which modern readers are for the determination of ultimate objective truth, and his familiar; under the name of airhuara, the three principles followers, if not himself, adopted the ancient word axiom of construction now called postulates, together with the for the principles which, with the help of the criterion, three theoretic principles, specially geometrical, now printed they proceeded freely to excogitate. About the same as the tenth, eleventh, and twelfth axioms; finally, under time the authority of all general principles began to be the name of Koival évvorai, or common notions, the series considered more explicitly in the light of their origin. of general assertions concerning equality and inequality, Not that ever such considerations had been wholly overhaving an application to discrete as well as continuous looked, for, on the contrary, Aristotle, in pronouncing the quantity, now printed as the first nine axioms. Now, principles of demonstration to be themselves indemonthroughout the Elements, there are numerous indications, strable, had suggested, however obscurely, a theory of that Euclid could not have been acquainted with the logical their development, and his followers, having obscure say. doctrines of Aristotle: a most important one has been ings to interpret, had been left free to take different sides signalized in the article ANALYSIS, and, in general, it may on the question; but, as undoubtedly the philosophic insuffice to point out that Euclid, who is said to have flour- vestigation of knowledge has in the modern period become ished at Alexandria from 323 (the year of Aristotle's death) more and more an inquiry into its genesis, it was inevitable to 283 B.C., lived too early to be affected by Aristotle's that principles claiming to be axiomatic should have their work—all the more that he was, by philosophical profes- pretensions scanned from this point of view with closer sion, a Platonist. Yet, although Euclid's disposition of vision than ever before. Locke it was who, when the geometrical principles at the beginning of his Elements is Cartesian movement was well advanced, more especially itself one among the signs of his ignorance of Aristotle's gave this direction to modern philosophic thought, turning logic, it would seem that he had in view a distinction be attention in particular upon the character of axioms; nor tween his postulates and common notions not unlike the was his original impulse weakened-rather it was greatly Aristotelian distinction between airhuara and áşubuara. strengthened-by his followers' substitution of positive All the postulates of Euclid (including the last three so- psychological research for his method of general criticism. called axioms) may be brought under Aristotle's descrip- The expressly critical inquiry undertaken by Kant, at tion of airhuara-principles concerning which the learner however different a level, had a like bearing on the ques. has, to begin with, neither belief nor disbelief (Post. Anal., tion as to the nature of axiomatic principles; and thus it i. 10,6); being (as De Morgan interprets Euclid's meaning) has come to pass that the chief philosophic' interest now such as the reader must grant or seek another system, attached to them turns upon the point whether or not they whatever be his opinion as to the propriety of the assump- have their origin in experience. tion.” Still closer to the Aristotelian conception of axioms It is maintained, on the one hand, that axioms, like come Euclid's common notions, as principles “which there other general propositions, result from an elaboration of is no question every one will grant” (De Morgan). From particular experiences, and that, if they possess an excepthis point of view, the composition of Euclid's two lists, as tional certainty, the ground of this is to be sought in the they originally stood, becomes intelligible: be this, how- character of the experiences, as that they are exceptionally ever, as it may, there is evidence that his enumeration and simple, frequent, and uniform. On the other hand, it is division of principles were very early subjected to criticism held that the special certainty, amounting, as it does, to by his followers with more or less reference to Aristotle's positive necessity, is what no experience, under any cirdoctrine. Apollonius (250–220 B.C.) is mentioned by Pro- cumstances, can explain, but is conditioned by the nature clus (Com. in Eucl., iii.) as having sought to give demonstra- of human reason. More it is hardly possible to assert genertions of the common notions under the name of axioms. ally concerning the position of the rival schools of thought, Further, according to Proclus, Geminus made the distinc for on each side the representative thinkers differ greatly tion between postulates and axioms which has become the in the details of their explanation, and there is, moreover, on both sides much difference of opinion as to the scope of determinately opposed since the time and mainly through the question. Thus Kant would limit the application of the influence of Kant. Such axioms, according to Kant the name axiom to principles of mathematical science, being necessary as well as synthetic, cannot be got from denying that in philosophy. (whether metaphysical or experience, but depend on the nature of the knowing datoral), which works with discursive concepts, not with faculty ; being immediately synthetic, they are not thought intuitions, there can be any principles immediately certain; discursively, but apprehended by way of direct intuition. end, as a matter of fact, it is to mathematical principles According to the experientialists, as represented by J. S. only that the name is universally accorded in the language Mill, they are, for all their certainty, inductive generaliof special science-not generally, in spite of Newton's lead, zations from particular experiences; only the experiences to the laws of motion, and hardly ever to scientific principles are peculiar (as already said) in being extremely simple of more special range like the atomic theory. Other think- and uniform, while the experience of space-Mill does not ers, however, notably Leibnitz, lay stress on the ultimate urge the like point as regards number—is farther to be principles of all thinking as the only true axioms, and would distinguished from common physical experience in that it contend for the possibility of reducing to these (with the supplies matter for induction no less in the imaginative belp of definitions) the special principles of mathematics, (representative) than in the presentative form. Mill thus commonly allowed to pass and do duty as axiomatic. Still agrees with Kant on a vital point in holding the axioms others apply the name equally and in the same sense to to be synthetic propositions, but takes little or no account the general principles of thought and to some principles of of that which, in Kant's eyes, is their distinctive characspecial science. In view of such differences of opinion as teristic—their validity as universal truths in the guise of to the actual matter in question, it is not to be expected direct intuitions or singular acts of perception, presentathat there should be agreement as to the marks character- tive or representative. The synthesis of subject and predistic of axioms, nor surprising that agreement, where it icate, thus universally valid though immediately effected, appears to exist, should often be only verbal. The charac- Kant explains by supposing the singular presentation or ter of necessity, for example, so much relied upon for ex- representation to be wholly determined from within cluding the possibility of an experiential origin, may through the mind's spontaneous act, instead of being reeither, as by Kant, be carefully limited to that which can ceived as sensible experience from without; to speak more be claimed for propositions that are at the same time syn- precisely, he refers the apprehension of quantity, whether thetic, or may be vaguely taken (as too frequently by continuous or discrete, to "productive imagination,” and Leibnitz) to cover necessity of mere logical implication regards it always as a pure mental construction.' Mill, the necessity of analytic, including identical, proposi- who supposes all experience alike to be passively received, tions—which Kant allowed to be quite consistent with or, at all events, makes no distinction in point of original origin in experience. The question being so perplexed, apprehension between quantity and physical qualities, fails no other course seems open than to try to determine the to explain what must be allowed as the specific character nature of axioms mainly upon such instances as are, at of mathematical axioms. Our conviction of their truth least practically, admitted by all, and these are mathemat- cannot be said to depend upon the amount of supporting ical principles.
experience, for increased experience (which is all that Mill That propositions with an exceptional character of cer- secures and secures only for figured magnitude, without tainty are assumed in mathematical science is notorious; psychological reason given) does not make it stronger; that such propositions must be assumed as principles of the and, if they are conceded on being merely stated, which, science, if it is to be at once general and demonstrative, is unless they are held to be analytic propositions, amounts now conceded even by extreme experientialists; while it is, to their being granted upon direct inspection of a particfarther, universally held that it is the exceptional character ular case, it can be only because the case, so decisive, is of the subject-matter of mathematics that renders possible made and not found-is constituted or constructed by such determinate assumption. What the actual principles ourselves, as Kant maintains, with the guarantee for unito be assumed are, has, indeed, always been more or less formity and adequacy which direct construction alone disputed; but this is a point of secondary importance, since gives. Still it does not therefore follow that the construcit is possible from different sets of assumption to arrive at tion whereby synthesis of subject and predicate is directly results practically the same. The particular list of propo- made is of the nature described by Kant—due to the sitions passing current in modern times as Euclid's axioms, activity of the pure ego, opposed to the very notion of like his original list of common notions, is open to objection, sensible experience, and absolutely a priori. As we have not so much for mixing up assertions not equally underiv- a natural psychological experience of sensations passively ative (as the ancient critics remarked), but for including received through bodily organs, we also have what is not two-the 8th and 9th—which are unlike all the others in less a natural psychological experience of motor activity being mere definitions (viz., of equals and of whole or part). exerted through the muscular system. Only by muscular Being intended as a body of principles of geometry in movements, of which we are conscious in the act of perparticular within the general science of mathematics, the forming them, have we perception of objects as extended modern list is not open to exception in that it adds to the and figured, and in itself the activity of the describing and propositions of general mathematical import, forming circumscribing movements is as much matter of experience Euclid's original list, others specially geometrical, pro- as is the accompanying content of passive sensation. At vided the additions made are sufficient for the purpose. It the same time, the conditions of the active exertion and does, in any case, contain what may be taken as good rep. of the passive affection are profoundly different. While, resentative instances of mathematical axioms both general in objective perception, within the same or similar moveand special; for example, the 1st, Things equal to the ments, the content of passive sensation may indefinitely same are equal to one another, applicable to all quantity; vary beyond any control of ours, it is at all times in our and the 10th, Two straight lines cannot enclose a space, power io describe forms by actual movement with or specially geometrical. (The latter has been regarded by without a content of sensation, still more by represented some writers as either a mere definition of straight lines, or imagined movement. Our knowledge of the physical or as contained by direct implication in the definition; but qualities of objects thus becomes a reproduction of our incorrectly. If it is held to be a definition, nothing is too manifold sensible experience, as this in its variety complex io be so called, and the very meaning of a defini- can alone be reproduced, by way of general concepts; tion as a principle of science is abandoned; while, if it is our knowledge of their mathematical attributes is, first held to be a logical implication of the definition, the whole and last, an act of conscious production or construction. science of geometry may as well be pronounced a con- It is manifestly so, as movement actual or imaginary, in geries of analytic propositions. When straight line is the case of magnitude or continuous quantity; nor is it strictly defined, the assertion is clearly seen to be syn- otherwise in the case of number or discrete quantity, when thetic.) Now of such propositions as the two just quoted the units are objects (points or anything else) standing it is commonly said that they are self-evident, that they apart from each other in space. When the units are not are seen to be true as soon as stated, that their opposites are objects presented to the senses or represented as coexistent inconceivabie; and the expressions are not too strong as in space, but are mere subjective occurrences succeeding descriptive of the peculiar certainty pertaining to them. each other in time, the numerical synthesis, doubtless, proNothing, however, is thereby settled as to the ground of ceeds differently, but it is still an act of construction, dethe certainty, which is the real point in dispute between pendent on the power we have of voluntarily determining the experiential and rational schools, as these have become the flow of subjective consciousness. Thus acting con