The main features of Egyptian mathematical science are then as follows: about 2000 B.C. a well-developed use of whole numbers and fractions; a method of solving equations of the first degree with one unknown quantity; an approximate method for finding the circumference of a circle of given radius; approximate methods for finding areas of isosceles triangles and trapezoids; the rudiments of a theory of similar figures.

REFERENCES FOR READING

BALL. A Short History of Mathematics, Chapter I. CAJORI. A History of Mathematics, pages 1-15. BERRY. A History of Astronomy, Chapter I. DREYER. Planetary Systems, Introduction. Gow. History of Greek Mathematics, Chapters I, II.

MAP OF THE WORLD BY HECATEUS (517 B.C.)

(From Breasted's Ancient Times. Courtesy of Messrs. Ginn & Co.)

Hecatæus, a geographer of Miletus, travelled widely, including a journey up the Nile, and wrote a geography of the world. In this book, as in the Map... the Mediterranean Sea was the centre and the lands about it were all those known to the author.... After the Unknown Historian of the Hebrews [about 850 B.C.] he was the first historical writer of the early world. -Breasted.

CHAPTER III

THE BEGINNINGS OF SCIENCE IN GREECE

Except the blind forces of Nature nothing moves in this world which is not Greek in its origin. -Sir Henry Sumner Maine.

A spirit breathed of old on Greece and gave birth to poets and thinkers. There remains in our classical education I know not what of the old Greek soul-something that makes us look ever upward. And this is more precious for the making of a man of science than the reading of many volumes of geometry. - Poincaré.

Number, the inducer of philosophies,

The synthesis of letters. - Eschylus.

Mathematics, considered as a science, owes its origin to the idealistic needs of the Greek philosophers, and not as fable has it, to the practical demands of Egyptian economics. . . Adam was no zoölogist when he gave names to the beasts of the field, nor were the Egyptian surveyors mathematicians. Hankel.

GEOGRAPHICAL BOUNDARIES. — From the twilight of civilization and the first faint suggestions of science in Chaldea and Egypt, we pass to the more brilliant dawn of science and civilization in Greece. Geographically we shall be concerned not merely with Greece itself, but, as time passes, with other Hellenic countries, especially the Ionian shores and islands of western Asia Minor, and the Greek colonies in southern Italy, Sicily, and, after its conquest by Alexander the Great, northern Egypt. Greece and its civilization seem immeasurably closer to us both in time and in spirit than do ancient Babylonia and Egypt. In these more remote civilizations science had been cultivated chiefly as a tool, either for immediate practical applications or as a part of the professional lore of a conservative priesthood. In Greece, on the other hand, for the first time in the history of our race,

human thought achieved freedom, and real science became possible.

Mathematics as a science commenced when first some one, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. -Whitehead.

INDEBTEDNESS OF GREECE TO BABYLONIA AND EGYPT. It is plain, nevertheless, that Greek civilization and Greek science owed much to Egypt and Chaldea. Herodotus has been quoted already, and Theon of Smyrna (second century A.D.) says:

In the study of the planetary movements the Egyptians had employed constructive methods and drawing, while the Chaldeans preferred to compute, and to these two nations the Greek astronomers owed the beginnings of their knowledge of the subject.

Again in the third century A.D. Porphyry observes:

From antiquity the Egyptians have occupied themselves with geometry, the Phoenicians with numbers and reckoning, the Chaldeans with theorems.

THE GREEK POINT OF VIEW. - It is not, however, so much the achievements of the Greeks in positive science which compel our attention and admiration as it is the remarkable spirit which they displayed toward man and the universe. Here for the first time we meet with a new point of view, and while Shelley's well-known dictum, "We are all Greeks, our laws, our literature, our religion, our art have their roots in Greece," must be dismissed as incorrect as well as extravagant, and even Sir Henry Maine's maxim, which stands at the head of this chapter, is undoubtedly an exaggeration, these famous sayings serve well to illustrate the fact that with the Greeks came into the world a new spirit and a new interpretation of Nature.

In a striking essay entitled "What we owe to Greece," Butcher has portrayed with extraordinary clearness those characteristics of the Greeks which lifted them above all of their predecessors and above most if not all of those that have come after them:

The Greeks before any other people of antiquity possessed the love of knowledge for its own sake. To see things as they really are, to discern their meaning and adjust their relations, was with them an instinct and a passion. Their method in science and philosophy might be very faulty and their conclusions often absurd, but they had that fearlessness of intellect which is the first condition of seeing truly. . . Greece, first smitten with the passion for truth, had the courage to put faith in reason and in following its guidance to take no account of consequences. "Those,' says Aristotle, 'who would rightly judge the truth must be arbitrators and not litigants.' 'Let us follow the argument wheresoever it leads' may be taken not only as the motto of the Platonic philosophy but as expressing one side of the Greek genius. . ..

At the moment when Greece has come into the main current of the world's history, we find a quickened and stirring sense of personality and a free people of intellectual imagination. The oppressive silence with which Nature and her unexplained forces had brooded over man is broken. Not that the Greek temper is irreverent or strips the universe of mystery. The mystery is still there and felt ... but the sense of mystery has not yet become mysticism. . . . Greek thinkers are not afraid lest they should be guilty of prying into hidden things of the gods. They hold frank companionship with thoughts that had paralyzed Eastern nations into dumbness or inactivity, and in their clear gaze there is no ignoble terror. Know thyself, is the answer which the Greek offers to the sphinx's riddle. . . . But to the Greeks, 'know thyself' meant not only to know man but the less pleasing task to know foreigners. . . . The people of ancient India did not care to venture beyond their mountain barriers and to know their neighbors. The Egyptians, though in certain branches of science they had made progress, in medicine, in geometry, in astronomy, had acquired no scientific distinction for they kept to themselves, but the Greeks were travellers. . . . Aristotle thought it worth his while to analyze and describe the constitutions of 58 states, including in his survey not only Greek states but those of the barbarian world.

It was the privilege of the Greeks to discover the sovereign efficacy of reason. . . And it was Ionia that gave birth to the idea which was foreign to the East but has become the starting-point of modern science, the idea that Nature works by fixed laws. . . . Again, in

history the Greeks were the first who combined science and art, reason and imagination. . . . The application of a clear and fearless intellect to every domain of life was one of the services rendered by Greece to the world. It was connected with an awakening of the lay spirit. In the East the priests had generally held the keys of knowledge. . . . To Greece then we owe the love of science, the love of art, the love of freedom. . . . And in this union we recognize the distinctive features of the West. The Greek genius is the European genius in its first and brightest bloom.

SOURCES. The sources of our information as to the details of the scientific ideas of the Greeks are exceedingly meagre, some of the most important historical and scientific treatises being known to us only by title or by detached quotations, or indirectly through Arabic translations. Among specific ancient sources of information in regard to Greek mathematical science the following may be mentioned:

About 330 B.C., Eudemus, a disciple of Aristotle, wrote a history of geometry of which a summary by Proclus has been preserved.

About 70 B.C., Geminus of Rhodes wrote an Arrangement of Mathematics with historical data. This has also been lost, but quotations are preserved in some of the later authors.

About 140 A.D., Theon of Smyrna wrote Mathematical Rules necessary for the Study of Plato.

About 300 A.D., Pappus' Collections contain much information in regard to the previous development of geometry.

In the fifth century A.D., Proclus published a commentary on Euclid's Elements with valuable historical data.

THE CALENDAR.The Greek calendar was based at an early period on the lunar month, the year consisting of 12 months of 30 days each. About 600 B.C. a correction was made by Solon, making every two years contain 13 months of 30 days and 12 of 29 days each, giving thus 369 days per year. In the following century a much closer approximation - 365 days- was attained by confining the thirteenth month to three years out of eight. This arrangement naturally failed, however, to meet the