Whether they are destined to remain merely monuments of the ingenuity and acute-ness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict. According to a tradition handed down by the Greek historians of geometry, the science took its rise among the Egyptians. The inundations of the Nile annually obliterated their landmarks, and efforts to restore them gave rise to geometry. From them, about 6OO B. C, Thales of Miletus, one of the seven wise men" of Greece, is said to have derived a knowledge of the elements of geometry, and to have introduced it into Greece. Pythagoras is also said to have derived his first notions of geometry from the same source, and to him is ascribed the discovery of the proposition, which still bears his name, that the square described on the hy-pothenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides. His disciples are said to have demonstrated the incommensurability of the diagonal and side of a square, and to have investigated the five regular solids.

They were also possibly acquainted with the transcendental definition of the circle, viz., that it is the figure which within a given perimeter contains the greatest area; and with the analogous proposition in regard to the sphere, that it is the body which within a given surface contains the greatest volume. About a century after Pythagoras, Plato and his disciples commenced a course of rapid and astonishing discoveries, through the study of the analytic method, conic sections, and geometric loci. The ancient analytic mode consisted in assuming the truth of the theorem to be proved, and then showing that this implied the truth only of those propositions which were already known to be true. In modern days the algebraic method, since it allows the introduction of unknown quantities as data for reasoning, has usurped the name of analytic. Conic sections embrace the study of the curves generated by intersecting a cone by a plane surface.

Within 150 years after Plato's time this study had been pushed by Apollonius and others to a degree which has scarcely been surpassed by any subsequent geometer, and his works, embracing his predecessors' discoveries as well as his own, proved 19 centuries afterward the foundation of a new system of astronomy and mathematics. Geometrical loci are lines or surfaces defined by the fact that every point in the line or the surface fulfils one and the same condition of position. The investigation of such loci has been from Plato's day to the present one of the most fruitful of all sources of geometrical knowledge. Just before the time of Apollonius, Euclid introduced into geometry a device of reasoning which was exceedingly useful in cases where neither synthesis (i. e., direct proof) nor the analytic mode is readily applicable; it consists in assuming the contrary of your proposition to be true, and then showing that this implies the truth of what is known to be false. Contemporary with Apollonius was Archimedes (died in 212 B. C), who introduced into geometry the fruitful idea of exhaustion.

By calculating circumscribed and inscribed polygons about a curve, and increasing the number of sides until the difference between the external and internal polygons becomes exceedingly small, it is evident that the difference between the curve and either polygon will be less than that between the polygons themselves; and the process may be continued by increasing the number of sides, until the difference between the curve and the polygon is as small as we please. This method is generally regarded as the germ of the differential calculus. Hipparchus in the 2d century before Christ, and Ptolemy in the 2d century after Christ, applied mathematics to astronomy; at the date of the hitter writer the doctrine of both plane and spherical triangles had been well discussed by Theodosius and Menelaus. Vieta (1540-1003), to whom we principally owe the perfecting of algebra, enlarged Plato's analytic method by applying algebra to geometry. Kepler (1571-1030) introduced into geometry the idea of the infinitesimal, thus perfecting the Archimedean exhaustion; he also first made the important remark which leads to the solution of questions of maxima, that when a quantity is at its highest point its rise becomes zero.

To Kepler we owe also one of the first examples of a problem of descriptive geometry, in the graphic solution of an eclipse of the sun. Soon after Kepler, Cavalieri published (1035) his Geometria Indivisibilibus, a further step in the road from Archimedes's exhaustions to Newton's fluxions. Roberval gave a method of drawing tangents identical in its philosophy with fluxions. Fermat (who shares with Pascal the credit of inventing the calculus of probabilities) introduced the infinitesimal into algebraical calculation, and applied it with great success to geometrical questions. Pascal anticipated some of the latest inventions by his famous theorem concerning the relation of six points arbitrarily chosen in a conic section. But most wonderful of all the geometrical inventions of the 17th century was that of Descartes, published in 1037; it consisted simply in considering every line as the locus of a point whose position is determined by a relation between its distances from two fixed lines at right angles to each other. The relation between these distances, being expressed in algebraical language, constitutes the equation of the curve.

By later geometers this method has been generalized so that the distances may be measured from any fixed point or line, and measured in a straight line or in a given curved line; or instead of some of the distances, directions or angles may be introduced. For a majority of the most important cases, however, Descartes's coordinates are still the best. Huygens, whose treatise on the pendulum is ranked by Chasles with Newton's Principia, making a combina--tion of Descartes's methods with those of his predecessors, added to geometry the beautiful theory of evolutes, which are the curves formed by the intersection of straight lines at right angles to a given curve; and he applied it not only to the pendulum, but to the theory of optics. Soon after (1686) Tschirnhausen published a wider conception of the generation of curves by straight lines. His famous caustics were made by the intersection of reflected or refracted rays of light; and he proposed other curves made by a pencil point stretching a thread whose ends were fastened, and which also wrapped and unwrapped from given curves. About the same time also De la Hire and Le Poivre invented, independently of each other, modes of transforming one plane curve into another, by making the given curve a peculiar basis for the locus of a new curve.