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his peer in Apollonius. . . . His work is only the last convulsive effort of Greek geometry, which was now nearly dead, and was never effectually revived. . . . It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations, indeed, but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green. But no Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life.

A melancholy interest attaches to the fate of Hypatia, daughter of Theon an Alexandrian mathematician, herself a teacher of Greek philosophy and mathematics, who was torn to pieces by a Christian mob, doubtless as a representative of pagan (Greek) learning, at Alexandria in 415 A.D.

CONCLUSION AND RETROSPECT. - Intellectual interests in the Greek world (now really Roman) were by this time so completely alienated from mathematics, and indeed from science in general, that the brilliant work of Pappus and Diophantus aroused but slight and temporary interest. Geometry had reached within the possible range of the Euclidean method a relatively complete development. Algebra under Diophantus attained in spite of hampering notation a level not again approached for many cen

turies.

Little need be said of sciences other than those already dealt with. These, even more than mathematics and astronomy, shrank under Roman autocracy and Christian hostility. Only the works of Galen, Strabo, and Pliny need be mentioned, and with them we deal in the next chapter.

The torch of science now passes from the Greeks to the Indians of the far East after their conquest by Alexander, to be in turn

surrendered to the Mohammedan conquerors of Alexandria A.D. 641. By them it is kept from extinction until in later ages it is once more fanned to ever increasing radiance in western Europe. In attempting a retrospective estimate of Greek science it is fundamentally important to judge the whole background fairly. In science the Greeks had to build from the foundations. Other peoples had extensive knowledge and highly developed arts. Only among the Greeks existed the true scientific method with its characteristics of free inquiry, rational interpretation, verification or rectification by systematic and repeated observation, and controlled deduction from accepted principles.

The Assyrians, Babylonians and Egyptians had certainly made great progress in the use of mechanical devices for moving heavy loads, in the construction of scales, and of pumps. Their measuring instruments were well developed, and acute observations were made, but of systematic, scientific investigation there is no evidence. The Greeks received many results and suggestions from Asia Minor, Mesopotamia, and Egypt, but their achievements are essentially their own. Wiedemann.

In asking ourselves why these extraordinary beginnings seemed after a time to lose their power of continued development, we must not forget the effect of external conditions. It is conceivable indeed that scientific progress should continue from age to age, through the genius of individual teachers and students, regardless of political and social conditions. Such, however, is not the historic fact. For progress in science men of genius are indispensable, but in no country or age have they alone been able to make science flourish under conditions so unfavorable as were those of the early centuries of the Christian era.

Greek science, however, did not "fail," learned and elaborate as are the explanations that have been given of its alleged failure. Under "the chill breath of Roman autocracy" its growth was indeed checked, its animation suspended, for a full thousand years. Then in the Renaissance it renewed its vitality and has ever since been advancing more and more magnificently. This is not to say

that criticisms as to the imperfections of the Greek scientific method are invalid, but rather to assert, as most critics must agree, that its merits outweighed its defects, and that the latter would not have proved disastrous but for the development of political, economic and military conditions under which the free Greek spirit could not continue its wonderful achievements.

BALL. Chapters IV, V.

REFERENCES for Reading

BERRY. Chapter II, Articles 37-54.
DREYER. Chapters VI-IX.

Gow. Chapters IV, VIII, IX, X.

HEATH. Diophantus of Alexandria. Aristarchus of Samos.

CHAPTER VII

THE ROMAN WORLD. THE DARK AGES

Among them [the Greeks] Geometry was held in highest honor: nothing was more glorious than Mathematics. But we have limited the usefulness of this art to measuring and calculating. - Cicero. The Romans were as arbitrary and loose in their ideas as the Greeks, without possessing their invention, acuteness and spirit of system. - Whewell.

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The Romans, with their limited peasant horizon and their shortsighted practical simplicity, cherished always for true science in their inmost hearts that peculiar mixture of suspicion and contempt which is so familiar today among the half educated. The arch dilettante Cicero boasts, even, that his countrymen, thank God! are not like those Greeks, but confine the study of mathematics and that sort of thing to the practically useful. - Heiberg.

THE ROMAN WORLD-EMPIRE. - For several centuries, during the decline of Greek learning both in Greece itself and in Alexandria, two new and powerful States were developing; one having its centre at Carthage on the northern shore of Africa, almost opposite Sicily, the other-the Roman Empire - on the western shore of Italy in the valley of the Tiber. The latter, at first comparatively insignificant, rapidly rose to a position of world-wide power, conquering in turn Carthage, Greece, and the East and eventually extending over the greater part of the then known world, from Britain on the north to the Cataracts of the Nile on the south, from India in the east to the Pillars of Hercules in the west.

THE ROMAN ATTITUDE TOWARDS SCIENCE. One of the most striking facts in the history of science is the total lack of any evidence of real interest in science or in scientific research among the Roman people itself or any people under Roman sway. Alexandrian science, even, though previously flourishing, languished and went steadily to its fall after the submission of that city to the Romans in the first century B.C. The truth seems to be that

the Roman people, while highly gifted in oratory, literature, and history (as witness, for example, the works of Cicero, Virgil and Tacitus), were not interested and therefore not successful in scientific work. This is the more impressive when we reflect upon their marvellous military genius, and their preeminence in world-wide power, dominion and influence. In vain do we look for any Roman scientist or philosopher of such originality or range as Aristotle or Plato; for any Roman astronomer, like Aristarchus or Hipparchus or Ptolemy; for any Roman mathematician or inventor, like Archimedes; for any Roman natural philosopher, like Democritus; for any Roman pioneer in medicine, like Hippocrates, — for Galen was Roman neither by birth nor education, but only by adoption late in life.

ROMAN ENGINEERING AND ARCHITECTURE. There is however one marked feature of Roman civilization in which extraordinary ability was displayed and peculiar excellence achieved and in which the Romans were unquestionably far superior to all their predecessors and, until very recent times, to all their successors. This feature, which is one of the most characteristic, is the Roman genius for both military and civil engineering. It is only necessary to mention the surviving remains of Roman walls, fortresses, roads, aqueducts, theatres, baths, and bridges. Never before and never since has any empire built so many, so splendid, and so enduring monuments for the service of its peoples in peace and in war. The surface of southern Europe, western Asia and northern Africa is still covered after the lapse of twenty centuries with Roman remains which bid fair to resist decay and destruction for another two thousand years. Roman engineering is almost as distinguished as is Roman law. The Emperor Constantine in the fourth century wrote: "We need as many engineers as possible. As there is lack of them, invite to this study persons of about 18 years, who have already studied the necessary sciences. Relieve the parents of taxes and grant the scholars sufficient means." The land surveyors formed a well-organized gild, but they were merely practitioners of a traditional art, perpetuating the errors of their ancient Egyptian predecessors, not dreaming of new dis

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