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 combination of arithmetic with geometry. The notions of an equation and a proportion-which are common to both, and contain the first germ of algebra-were introduced among the Greeks by Thales. These notions, especially the latter, were elaborated by Pythagoras and his school, so that they reached the rank of a true scientific method in their theory of proportion. To Pythagoras, then, is due the honour of having supplied a method which is common to all branches of mathematics, and in this respect he is fully comparable to Descartes, to whom we owe the decisive combination of algebra with geometry.

See G. J. Allman, Greek Geometry from Thales to Euclid (Cambridge, 1889); M. Cantor, Vorlesurzgen iiber Geschichte der Mathematik Leipzig, 1894); James Gow, Short History of Greek Mathematics Cambridge, 1884). (G. J. A.)

PYTHAGORAS, of Rhegium, a noted Greek sculptor of the 5th century B.c., a contemporary of Myron and Polyclitus, and their rival in making statues of athletes. He was born at Samos and migrated in his youth to Rhegium in Italy. He made a statue of Philoctetes notable for the physical expression of pain, an Apollo shooting the Python at Delphi, and a man singing to the lyre. He is said to have introduced improvements in the rendering of muscles, veins and hair.

PYTHEAS, of Marseilles (Massilia), a celebrated Greek navigator and geographer, from whom the Greeks apparently derived their earliest definite information concerning western Europe, and especially the British Islands. He was probably contemporary with Alexander the Great; he certainly wrote before Dicaearchus, a pupil of Aristotle who died about 285 B.C. His work is lost, and we are left almost wholly in the dark as to its form and character, but the various titles under which it is quoted (e.g. Γῦς περίοδος, or Τἀ περὶ τοῦ Ὠκεανόῦ) point to a geographical treatise, in which Pytheas had embodied the results of his observations, rather than to a continuous narrative of his voyage.

Some modern writers have supposed Pytheas to have been sent out, at public expense, in command of an expedition organized by the republic of Massilia; but there is no ancient authority for this, and Polybius, who had unquestionably seen the original work, expressly states that he had undertaken the voyage in a private capacity and with limited means. All that we know concerning the voyage of Pytheas (apart from detached notices) is contained in a brief passage of Polybius, cited by Strabo, in which he tells us that Pytheas, according to his own statement, had not only visited Britain, but had personally explored a large part of it (“ travelled all over it on foot,” according to one reading of the text in Strabo, bk. iv. ch. i.), and estimated its circumference at more than 40,000 stadia (4000 geographical miles). To this he added the account of Thule (which he placed six days' voyage north of Britain) and the adjoining regions, in which there was no longer any distinction between air, earth and sea, but a kind of mixture of all three, resembling the gelatinous mollusc known as pulmo marinus, which rendered all navigation and progress in any other mode alike impossible. This substance Pytheas had himself seen, according to Strabo (bk. iv. ch. i.), but the other phenomena he described only from hearsay. After this he visited “ the whole of the coasts of Europe ” (i.e. those bordering on the ocean) as far as the Tanais (Strabo, bk. ii. ch. iv. § 1). This last sentence has led some modern writers to suppose that he made two different voyages; but this is improbable; the expressions of Polybius imply that his explorations in both directions, first towards the north and afterwards towards the east, formed part of the same voyage.

The countries visited, and to a certain extent explored, by Pytheas, were previously, unknown to the Greeks—except, perhaps, by vague accounts received through the Phoenicians—and were not visited by any subsequent authority during more than two centuries. Hence some of the later Greek geographers altogether disregarded his statements, and treated his voyage as a fiction. Eratosthenes, indeed (276–196 B.c.), attached great value to his authority as to Britain and Spain, though doubting some of his statements; but Polybius (c. 204–122 B.C.) considered the whole work of Pytheas a tissue of fables, like that of Euhemerus concerning Panchaea; and even Strabo, in whose time the western regions of Europe were comparatively well known, adopted to a great extent the view of Polybius.

In modern times a critical examination has arrived at a more favourable judgment, and though Gossellin in his Recherches sur la géographie des anciens (iv. 168–180) and Sir G. C. Lewis in his History of Ancient Astronomy (pp. 466–481) revived the sceptical view, the tendency of modern critics has been rather to exaggerate than to depreciate the value of what was really added by Pytheas to knowledge. Our information concerning him is so imperfect, and the scanty notices preserved to us from his work are so meagre and discordant, that it is difficult to arrive at anything like a sound conclusion. It may, however, be considered as fairly established that Pytheas made a voyage round the western coasts of Europe, proceeding from Gades, the great Phoenician emporium, and probably the farthest point familiar to the Greeks, round Spain and Gaul to the British Islands, and that he followed the eastern coast of Britain for a considerable, distance to the north, obtaining information as to its farther extension in that direction which led him greatly to exaggerate its size. At the same time he heard vaguely of the existence of a large island to the north of it—probably derived from the fact of the Orkneys and Shetlands being really found in that position—to which he gave the name of Thule.

The most important statement made by Pytheas in regard to Thule was that connected with the astronomical phenomena affecting the duration of day and night therein. Unfortunately the reports transmitted to us differ so widely that it is almost impossible to determine what Pytheas himself stated. It is, however, probable that the version given in one passage by Pliny (H. N. iv. 16, 104) correctly represents his authority. According to this, the days at the summer solstice were twenty-four hours in length, and conversely at the winter solstice the nights were of equal duration. Of course this would be true had Thule been situated under the Arctic Circle, which Pytheas evidently considered it to be, and his skill as an astronomer would lead him to accept as a fact what he knew must be true at some point as a voyager proceeded onwards towards the north.

Still more difficult is it to determine the extent and character of Pytheas's explorations towards the east. The statement that he proceeded along the coasts of Europe “ from Gades to the Tanais ” is evidently based upon the supposition that this would be a simple and direct course along the northern shores of Germany and Scythia—Polybius himself, in common with the other Greek geographers till a much later period, being ignorant of the projection of the Danish or Cimbric peninsula, and the circumnavigation that it involved—of all which no trace is found in the extant notices of Pytheas. Notwithstanding this, some modern writers have supposed him to have entered the Baltic and penetrated as far as the Vistula (his Tanais). The only foundation for this is to be found in the fact that in a passage cited by Pliny (H. N. xxxvii. 2, 35) Pytheas is represented as stating that amber was brought from an islhnd called Abalus, distant a day's voyage from the land of the Guttones, a German nation who dwelt on an estuary of the ocean called Mentonomus, 6000 stadia in extent. It was a production thrown up by the waves of the sea, and was used by the inhabitants to burn instead of wood. It has been conjectured that the “ estuary ” here mentioned refers to the Baltic, the existence of which as a separate sea was unknown to all ancient geographers; but the obscure manner in which it is indicated, as well as the inaccuracy of the statements concerning the place from whence the amber was actually derived, both point to the sort of hearsay accounts which Pytheas might readily have picked up on the shores of the German Ocean, without proceeding farther than the mouth of the Ems, Weser or Elbe, which last is supposed by Ukert to have been the limit of his voyage in this direction. It must be observed also that amber is found in Friesland and on the west coast of Schleswig, as well as in the Baltic, though not in equal abundance.

As to the Cassiterides, or Tin Islands, the exploration of which would naturally be one of the chief objects of Pytheas, he seems to have furnished Timaeus, who wrote less than a century after him, with details upon the same, especially in regard to the commercial centre of Iktis (St Michael's Mount in Cornwall ?), which are preserved by Diodorus. The trade with these regions was probably at this period in Phoenician hands, but we know that at a later time a considerable portion of the supply was carried overland through Gaul to Massilia.

Pytheas certainly had one merit which distinguished him from almost all his contemporaries—he was a good astronomer, and was one of the first who made observations for the determination of latitudes, among others that of his native place Massilia, which he fixed with remarlrable accuracy; his result, which was within a few miles of the truth, was adopted by Ptolemy, and became the basis of the Ptolemaic map of the western Mediterranean. His calculations of the length of the longest day at four different points in the neighbourhood of Britain are probably based on native reports. If these figures (16, 17, 18 and 19 hours) are to be pressed, they would refer to, say, Ushant (48° N.), Flamborough Head (54º), Tarbet Ness in Ross (58°) an the northernmost Shetlands (61°). Pytheas was also the first among the Greeks who arrived at any correct notion of the tides, and not only indicated their connexion with the moon, but pointed out their periodical fluctuations in accordance with the phases of that luminary. Other observations concerning the manners and customs of the inhabitants