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

Greek desire that the months begin regularly at or near new moon, and Aristophanes makes the Moon complain:

CHORUS OF CLOUDS

"The Moon by us to you her greeting sends,
But bids us say that she's an ill-used moon,
And takes it much amiss that you should still
Shuffle her days, and turn them topsy-turvy;
And that the gods (who know their feast-days well,)
By your false count are sent home supperless,
And scold and storm at her for your neglect."

About 400 B.C., Meton the Athenian observed that 19 years consist of almost exactly 235 lunar months, and accordingly proposed a new calendar with 125 months of 30 days and 110 of 29 days, corresponding to an average year of 365 days, 6 hours and 19 minutes only about 30 minutes too long. Of this Meton's cycle the traditional rule for determining the date of Easter still preserves traces. On account of so much confusion in the official calendar the almanacs of the time even designated the dates for agricultural operations by means of the constellations visible at the corresponding time.

TIME MEASUREMENT. While sun and moon suffice for largescale measurement of time, the approximate determination of its subdivisions early became important, and this problem has been solved with continually increasing precision to our own day. Early time measurement depended either on some form of sundial as a natural means, or on an apparatus analogous to the hour-glass as an artificial method.

In Isaiah xxxviii. 8, in connection with a promise of prolonged life to Hezekiah, it is said

And this shall be a sign unto thee from the Lord, that the Lord will do this thing that he hath spoken; behold, I will bring again the shadow of the degrees, which is gone down in the sun-dial of Ahaz, ten degrees backward. So the sun returned ten degrees, by which degrees it was gone down.

The first sun-dial of which a description is preserved belongs to the time of Alexander the Great, and consisted of a hollow hemisphere with its rim horizontal and a bead at the centre to cast the shadow. Curves drawn on the concave interior divided the period from sunrise to sunset into twelve parts, these lengths being thus proportionate to the lengths of the daylight period.

The use of the clepsydra, or water clock, in Greece dates from the fifth century B.C. It consisted there of a spherical bottle with a minute outlet for the gradual escape of water. Its use in regulating public speaking is illustrated by Demosthenes' demand when interrupted, "You there: stop the water."

For the sake of conformity with the sun-dial division of each day and each night into twelve equal parts, the rate of flow in the clepsydra required continual adjustment. Ingenious improvements were made in the mechanism in course of time, but in considering the work of the Greek astronomers, the impossibility of what we should consider accurate time measurement must not be forgotten.

GREEK ARITHMETIC. — In Greek arithmetic the earliest known numerals are merely the initials of the respective number words. Two other systems came into use later. In one of these the numbers from 1 to 24 are represented by the 24 letters of the Ionian alphabet; in the other the letters represent numbers, but no longer in consecutive order. This use of letters for numbers was not confined to Greece, but appears to have originated there. The Greeks had no zero, and never discovered the immense advantage of a position-system, such as that by which we are able to express all numbers by only ten symbols. Fractions occur not infrequently. The change from the earlier notation to that with 24 characters was a disastrous one. There were not only more characters to memorize, but computation became materially more complicated. These disadvantages far more than offset the superior compactness, the sole merit of the new system. The special importance of such compactness for coins has led to the suggestion that they were the medium through which this notation was introduced.

A simple numerical computation of late date in the Greek alphabetic numerals and its modern equivalent are

Division was an exceedingly laborious process of repeated subtraction.

Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. Whitehead.

Approximate square roots were found by the later Greeks. Theon in the fourth century A.D. for example gives the following rule: —

When we seek a square-root, we take first the root of the nearest square-number. We then double this and divide with it the remainder reduced to minutes and subtract the square of the quotient, then we reduce the remainder to seconds and divide by twice the degrees and minutes (of the whole quotient). We thus obtain nearly the root of the quadratic.

The reckoning board, or abacus, -known in so many different forms throughout the world, came into very early use, but actual evidence in regard to its form is meagre. A sharp distinction was made between the art of calculation (logistica), and the science of numbers (arithmetica). The former was deemed unworthy the attention of philosophers, and to their attitude may be fairly attributed the fact that Greek mathematics was always

weak on the analytical side, and seemed in a few centuries to reach the limit of its possible development.

GREEK GEOMETRY. It was in geometry that Greek mathematics chiefly developed, and for several fundamental reasons. The Greek mind had a strong predilection for formal logic, a keen æsthetic appreciation of beauty of form, and, on the other hand, with no adequate symbolism for arithmetic or algebra, a distinct disdain, at any rate among the educated, for the commercialized mathematics of computation. The history of Greek mathematics is therefore to a great extent the history of geometry. Formal geometry as distinguished from the solving of particular geometrical problems, had, indeed, no previous existence, and we have to do with the beginnings of elementary geometry as we now know it.

THE IONIAN PHILOSOPHERS. The sense of curiosity, the feeling of wonder, the spirit of inquiry, - these are the common elements of philosophy and science. It is thus not strange that the earliest names in science are likewise the earliest in philosophy.

In the childhood and youth of the race specialization has not begun, all knowledge lies invitingly open to the expanding mind. We have seen how much had been accumulated in Egypt and Babylonia of knowledge and skill in observing and recording the phenomena of the heavens, in irrigation and in measurement of land. Much of the same general character was doubtless true of the Phoenicians, the Trojans, the Cretans, and other precursors of the Greeks. But nothing deserving the name of science has come down to us from the Ægean or Greek civilization before the time of Thales of Miletus, chief of the Ionian philosophers, and one of the seven wise men of Greece."

THALES. The ancient and fragmentary register of Greek mathematicians, or history of Greek geometry before Euclid, attributed to Eudemus, begins:

As it is now necessary to consider also the beginnings of the arts and sciences in the present period, we report that, according to the evidence of most, geometry was invented by the Egyptians, taking its origin from the measurement of land. This last was necessary