GEOMETRY 701 covery 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 tvo sides. His disciples are said to have de- monstrated the incommensurability of the di- agonal and side of a square, and to have in- vestigated the five regular solids. They were also possibly acquainted with the transcen- dental definition of the circle, viz., that it is the figure which within a given perimeter con- tains the greatest area; and with the analo- gous proposition in regard to the sphere, that it is the body which within a given surface contains the greatest volume. About a cen- tury after Pythagoras, Plato and his disciples commenced a course of rapid and astonish- ing discoveries, through the study of the analy- tic method, conic sections, and geometric loci. The ancient analytic mode consisted in as- suming 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 alge"- braic method, since it allows the introduction of unknown quantities as data for reasoning, has usurped the name of analytic. Conic sec- tions embrace the study of the curves genera- ted 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, em- bracing 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 as- suming the contrary of your proposition to be true, and then showing that this implies the truth of what is known to be false. Con- temporary with Apollonius was Archimedes (died in 212 B. C.), who introduced into geom- etry the fruitful idea of exhaustion. By calcula- ting 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 mathe- matics to astronomy ; at the date of the latter writer the doctrine of both plane and spherical triangles had been well discussed' by Theodosius and Menelaus. Vieta (1540-1603), to whom we principally owe the perfecting of algebra, enlarged Plato's analytic method by applying algebra to geometry. Kepler (1571-1630) in- troduced into geometry the idea of the infini- tesimal, thus perfecting the Archimedean ex- haustion ; he also first made the important re- mark which leads to the solution of questions of maxima, that when a quantity is at its high- est point its rise becomes zero. To Kepler we owe also one of the first examples of a problem of descriptive geometry, in the graphic solu- tion of an eclipse of the sun. Soon after Kep- ler, Cavalieri published (1635) his Geometria Indivisibilibus, a further step in the road from Archimedes's exhaustions to Newton's flux- ions. Eoberval gave a method of drawing tangents identical in its philosophy with flux- ions. Fermat (who shares with Pascal the cred- it 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, pub- lished in 1637; it consisted simply in consid- ering 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 lan- guage, constitutes the equation of the curve. By later geometers this method has been gen- eralized 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 pub- lished a wider conception of the generation of curves by straight lines. His famous caus- tics 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,
