Evidently one plane cutting another will form a straight line; parallelism between planes will be determined by parallelism of the lines lying in those planes; and solid angles will be governed in formation by the plane angles which contain them. In figures bounded by right lines the added dimension, as in the instance of the cube, becomes a multiple merely of the two dimensions, as of a square, lying in one plane, whilst the determination of position is had by reference to the perpendicular, which makes right angles with every line drawn through its foot in the plane on which it stands. Parallels and perpendicular planes are determined and discussed by reference to the straight lines lying therein. This renders extremely simple the propositions touching magnitudes so bounded. Their general designation is that of polyhedrons, embracing pyramid, prism and parellopiped. The volume of the former equals base into one-third altitude, of the two latter, base into altitude.

And here it will be interesting to note a new method of illustrating solid geometry, which promises to add greatly to its beauty and perspicuity by presenting problems and diagrams stereoscopically. In this manner all the facility of the actual model may be had from the printed page or card, so that planes and angles and surfaces shall stand out in relief before the student, and assume, as in a magic mirror, their proper perspective.

Of figures bounded by curved surfaces, only the three round bodies are considered, they being generated respectively by triangle, rectangle, and circle, and designated cone, cylinder and sphere. In the matter of volume and surface, it will be at once apparent that they must sustain close relation, as they may have the same circle as an element. Without adverting to the methods of proof, it will suffice to say that, as to surface, the lateral area of the cone of revolution is the product of the circumference of its base by half its slant height; that of the cylinder is the product of a perimeter of a right section by an element of the surface; that of the sphere is the product of its diameter by the circumference of a great circle. The volume on the other hand of the cone is equal to one-third of base into the altitude; of the cylinder, equals base into altitude; of the sphere, equals the rea of its surface into one-third its radius.

And now, before proceeding to examine the more modern, and I may add the more highly developed stages of geometrical attainment, it may not be amiss to consider somewhat at large the annals which mark out the life and movement of so remarkable a sphere of thought amongst men.

Historically speaking, the science of geometry dates back to the early ages of Greek civilization. It was the favorite study of her sages from the time of Solon to that of Hipparchus, and was cultivated with an ardor scarcely now to be realized. Neither was it a thing apart from the ordinary occupations and interests of life, engaging merely the attention of the scholar, but constituted rather the moving force of the thought of that time, and thus permeated every phase of society. To the young and freeborn it afforded needful disciplines of severe study; to orator and statesman it gave a directness of speech and action which has never been surpassed; whilst to the philosophers and sophists it supplied those ample fields of contemplation so congenial and characteristic of the era. Nor did it conflict with any other dominant passion, for the singularly acute and versatile Ionic genius, which was destined to make the name of Athens so famous throughout the world, had turned from the founding of colonies to the enjoyments of leisure, and incited by such intercourse to a spirit of free inquiry, rushed with enthusiasm into paths of investigation offering such alluring prospect to the explorer. And there was a profound reason for this. The Greek life was all outdoors. The sky, the sea, the earth, fed the sensitive retina with fair shapes to delight and animate its play, until love of beauty became a greed, and its accurate portrayal the highest source of popular enjoyment. To wed its fleeting forms imperishably to marble, to master the secret of its curvature and grace, to model its proportion into temple or triumphal arch, to minister to the eye as the most exalted of all the senses, was a peculiarity reserved specially for the Hellenic race. It was thus therefore, as the very logic of luxury, that geometry, with its cultus of linear relation, insinuated itself into the socialism of that age and people. And the recompense was equal to the aspiration, for a whole community stood ready to crown with immortal honor those who might add to the heritage of such knowledge. No wonder Thales

poured out a great libation when he had demonstrated that the angles in a semicircle were all right angles! No wonder Pythagoras sacrificed a hecatomb of oxen to the gods when he discovered the square of the hypothenuse to be equal to the sum of the squares of the two other sides. No wonder Plato inscribed above the doorway of his sanctuary, "Let no one ignorant of geometry enter here," since he made it the basis of his entire system, thereby imposing it forever on the intellect of mankind; for how little of modern thought is there free of his infinite genius. A lifetime was all too short to expend in this behalf, when to found a school, to become a teacher, was the supreme ambition. And such were in reality the prizes for which men struggled in those days, so that the largest minds became devoted to sublime meditation upon time and space and quantity, feasting, as upon manna dropped from the supernal. Socrates would pass a whole summer's day by the banks of the bright Ilissus, debating the properties of numbers, whilst Anaxagoras employed his time in prison in striving to solve the quadrature of the circle. It was in fact the exaltation of a people suddenly awakened to intellectual life and the exuberance of spirit which attended the pursuit of such philosophies, was in no sense inferior to that which in a ruder age had given intense character to a siege of Troy, or Isthmian games, or an Amphyctionic League.

Extrinsic influences likewise operated in no slight degree to favor the same result. The defeat of Xerxes, the union of the states of Greece, and the splendid conquests of Alexander the Great, had changed entirely the economies of home-life. Suddenly there was placed untold wealth at the command of a people favored by long culture for its best uses, and they proceeded to build cities, to embellish temples, to establish libraries, to cultivate art, and above all, to give munificent encouragement to science. Thus it was to the plundered wealth of Darius that Aristotle was indebted for means to compile his works on natural history, whilst the gathered spoils of Arbela, jewels torn from the diadems of princes, and the ransom of beautiful women, made rich the sites of learning about all the shores of the Mediterranean. The world, however, has united in ascribing to Ptolemy Soter, a successor of the great conqueror, the proud distinction

of having first recognized and formulated the new spirit of the age. Egypt and its adjacent provinces had fallen to his share in the dismemberment which succeeded the Persian subjugation, and along with them the newly founded city of Alexandria, as a legacy to be made worthy the memory of the great name it bore. Perceiving the intellectual yearning which had culminated so rapidly, he sought to give it expression in the upbuilding there of another pyramid - greatest of them all. It was not to be, however, of huge blocks of granite, quarried by innumerable slaves through long years of toil; but gathering all the knowledge of the Orient into one immense repository, he purposed upon such spacious foundations to erect an edifice of learning that should be the cynosure of all time. In this far-reaching conception he was nobly sustained by his son and successor, and to their joint endeavor we may credit what was certainly the most brilliant age of scientific and especially mathematical culture in all past history. To the honor of letters, too, it must be said, they have ever held in gratitude these earliest patrons and dealt tenderly with the memory of the Ptolemies.

The imperial city, thus destined to bear so important a relation to the diffusion of knowledge, was built on a narrow neck of land separating Lake Mareotis from the Mediterranean. For its adornment the resources of a mighty kingdom, during two generations of men, were put in requisition. Marble and porphyry, temples and obelisks, spacious streets and tapestried bazaars, the pure white Pharos, and the golden coffin of him of Macedon, made memorable afar its many splendors. Four thousand palaces, gay with particolored stones, baths unsurpassed in extent or magnificence, and luxurious tropic markets, attested its heaped up riches. Near by flowed forth the deep silent river, with mystery written on its bosom, bearing the fate of empires in its rise and fall, and shrouding in its waters vanished civilizations which were great and proud when the world was young. Situated at the convergence of three continents, its population of six hundred thousand souls was made up from many countries, speaking many tongues. Pliant Greeks fresh from the schools of Sicily and of Rhodes, Hebrews trained in the teachings of the synagogue, Sarbeans and Phoenicians

and keen-witted Asiatics from the further shore of the Propontis, all struggled for preeminence. The moral atmosphere, too, was charged with metamorphosis. A decline of the oracles which had long held sway over the imagination of men, invited to a search after new gods. Astronomical observations, just beginning to yield to instrumental interpretation, gave entrancing interest to the mapping out of the heavens. Egypt likewise contributed her solemn centuries of meditation, whilst Hindoo ideas, creeping along the threads of commerce, accompanied Indian silks and Persian carpets to the renowned site at the mouth of the Nile. Splendid endowments gave the necessary support to a vast educational establishment, and distinguished scholars were maintained at the expense of the state. Students too, became free of the academies without reference to race or creed or sex. It was a carnival of speculative thought, with no ban on any knowledge to make it occult, and no premium on any faith to make it hypocritical. And so it came to pass that science in the only form then known, that of geometry, became predominant in the intellectual world by virtue of the very diversities in all other conviction. It was repose and belief both to the great awakening then abroad in the East.

The immediate fruit of this new garden of the Hesperides was, so to speak, apples of gold in pictures of silver. Here, upon the page of history, appear the names of the great geometers who may be considered the founders of the science. What preceded them had been isolated discovery; what followed was connected development. Let me dwell for a moment on some of the more renowned scholars who gave luster to this age.

First and foremost, in place as in time, stands Euclid, whose fame has held the reverence of all thinking men for more than two thousand years. Of his personal history very little is known, other than that he was taught by Aristasus, a pupil of Plato, and that he founded the school of mathematics at Alexandria during the reign of the first Ptolemy. He is described as mild in manner, unpretending in assertion, and full of kindliness to associates. His writings, however, have made his name familiar as a household word; for his "Elements", in which he gathered all that was then known of geometry, still form the most finished treatise on