tained by the whole line and one of the parts shall be equal to the square of the other part. As the very conception of a rectangle and a square involves a reference to parallel lines, our remark in the preceding paragraph is equally applicable to this. (6) That a triangle having the diameter of a circle as its base, and having its vertex in the circumference, is a rightangled triangle; in other words, that the angle in a semicircle is a right angle. (7) That an equilateral triangle, a square, a regular hexagon (probably also a regular pentagon), can be inscribed in a circle. (8) A doctrine of proportion, which referred to the distinction between commensurable and incommensurable quantities. Another important proposition, one case of which-but that the simplest-is used by Hippocrates. The proposition is that the circle described on the hypotenuse of a right-angled triangle as diameter is equal to the circles described on its sides. To us it seems that no one could know this without knowing also what is now Euc. VI. 31, and that Hippocrates inferred it from the analogy of the circle to the rectilineal figure.

For reasons which will afterwards appear, we restrict our summary to the matter contained in the first six books of the Elements. With solid geometry, as taught in Books XI. and XII., we do not concern ourselves here, though one part of it-that respecting the five regular solids-received much attention at the hands of geometers before Euclid.1

1 What are called Books XIV. and XV. of the Elements treat largely of these solids. But there is no reason to believe that Euclid was their author. They are now generally ascribed to Hypsicles, who was generally believed to have lived in the second century of our era, but whom Professor De Morgan, with apparently good reason, assigns to the sixth.

VI

"AT Babylon, 21st April, Alexander III., king of Macedonia, in the thirty-third year of his age and the thirteenth year of his reign." Such a notice might have appeared in the death column of the Macedon Times of 22nd April 323 B.C. An extra-Macedonian reader-say a Londoner-on reading this notice, would not have regarded the event so chronicled as one of very great importance, since Macedonia was, at the best, but a third-rate power, as it had had twenty-one kings, not one of whom, with the exception of Philip II., father of this Alexander, was ever heard of beyond the bounds of the small kingdom. The supposed reader, when he saw that the paper was broadly black-bordered, and contained a "leader," two columns long, setting forth that this Alexander had subdued not only neighbouring European provinces, but had waged brilliant and successful war against Persian and Syrian and Indian and Egyptian rulers,-had, in fact, created a "larger Macedonia" in Europe and Asia and Africa,— would doubtless have perceived that this hero's life was an important event in human history. But he would scarcely have realised that the death of this man was to mark one of the most signal epochs in the political and intellectual history of the world. The partition of his empire into four kingdoms, the mutual

wars and treaties between these kingdoms, their perpetual combinations and divisions, recombinations and redivisions, and their subjugation in succession by alldevouring Rome, constitute the staple of the world's history for many centuries. With these wars and tumults, amalgamations and separations, we have nothing to do; nor with any of the sections of the Alexandrian empire save one-the African; nor with that one, save in respect of an event in its early history, which at the time of its occurrence probably not regarded as of much importance.

66

The kingdom of Egypt, with no very definite boundaries, and apparently including, along with Egypt proper, Libya to the west and a portion of Arabia to the east, fell to the lot of Ptolemy, who is generally supposed to have been an illegitimate son of Philip, and therefore step-brother of Alexander. Be this as it may, he was certainly a special favourite of the father, and one of the bravest and most distinguished generals in the army of the son. It had been a hobby" of Alexander to erect in the lands which he conquered monuments to himself, in the form of cities called by his name. Thus in Asia there was a multitude of Alexandrias, no one of which had any lengthened existence, or left any history behind it. It was otherwise with the Egyptian Alexandria. Built on one of the finest maritime sites in the world, it soon became a most noted seaport, so that in old times it was regarded as the great granary of Rome. It has been the scene of most eventful contests in ancient and modern times; and although in our day the construction of the Suez Canal has materially diminished its importance, it continues, and in all probability will long

continue, to be a great commercial city, and the capital of what was once, and may be again, a great kingdom, whether independent or affiliated to a great empire.

The early years of the reign of the first Ptolemy were spent in wars, in which, it must be admitted, he seems generally to have been the aggressor, or at least, when there were faults on both sides, to be entitled to a full moiety of the blame. He found in his capital a variety of nationalities. We know that even in the time of Alexander a great number of Jews had settled as merchants, traders, and mechanics. Their religion, as with their fathers who dwelt in the same Egyptian land some two thousand years earlier, debarred them from amalgamation with the people of the land, or from other than commercial relations with them. They therefore became, and long continued to be, a separate community, and were the main components of that important body whom their countrymen in Palestine designated as Hellenists. With Ptolemy there came, of course, a large body of virtually Greek soldiers and officials. And as we have seen that from a much earlier time there were many Grecian visitors to Egypt, we may be sure that the number was not diminished when the country came under European rule. The Greeks were not, like the Jews, prevented by their religion from entering into any relations with the native Egyptians to which interest or inclination might prompt them. They were the dominant race, and the manners and habits of thought and life of such of the natives as came in contact with them were naturally assimilated to theirs; and so, though Egypt did not in any sense become a Grecian land, Alexandria soon became virtually a Grecian city.

When Ptolemy was fairly settled on his throne, he set himself vigorously to the device of schemes for elevating his subjects and advancing the glory of his capital. Having in his early days shared with his putative brother in the instructions of Aristotle, he could not fail to have vividly apprehended to what extent the cultivation of science and philosophy would contribute to these ends. He therefore determined to found a great school,-university, we should call it now, -with a library befitting the foreseen glory of his kingdom. The two together were called the MUSEUM. It is with the school that we have now to do; of the library we shall have to speak afterwards. The design was formed about 306 B.C., and by 300 B.C. the institution, with magnificent buildings and rich endowments, and a staff of distinguished teachers, was ready to enter on its illustrious career. We have seen that for centuries, from Thales to Plato, Greek after Greek went to Egypt to learn; now for the first time, so far as appears, they went to teach. Among those who accepted the invitation of Ptolemy was Euclid, who apparently in the year named-the first of the third century B.C.1 was installed as the first professor of mathematics in the University of Alexandria. At this time he was about thirty years old, and he occupied the position to which he was now called for twenty-five years, until his death in 275 B.C.

There seems to be no possibility of learning anything as to the date of his mathematical works, or ascertaining whether they were of Athenian or Alexandrian birth, or whether some were of the one and some of

1 So we maintain, just as we maintain that this year in which we write (1901) is the second, not the first, of the twentieth century of our era. The

cased are not an atogius,