GEOMETRY (Lat. geometria, from Gk. Aerpla geometria, from -yecoul-rns, gametres, geometer, from -yij, ye, earth + Fhb-pop, metron, measure). The science of form. Geometric con cepts arise from the consideration of forms of actual objects, just as numerical concepts arise from the consideration of collections of objects. E.g. the idea of a cube results from observing that the corresponding physical object, as a die occupies a certain part of space. This implies the first geometric assumption, viz. that space is divisible. In this case it is divided into two parts, that within the cube and that outside of it. Geometry considers only the former, the space occupied by a substance. This space is called a geometric solid or simply a solid. The boundary between the space and that outside of it is a surface. A surface, being itself an ele ment of space, is also divisible, and the boundary between two parts of it is called a line. A line, in turn, is divisible by a point. The number, comparative size, and position of these elements unite to make the concept cube. With accurate ideas of point, line, surface, solid, it is easy to imagine a world of geometric figures formed by their combinations. It is then only necessary to add concise definitions and axioms (q.v.) to found a system of geometry. But the validity of these assumed premises must determine the validity and scope of the resulting science—a fact forcibly exemplified in the case of Euclidean geometry.
Geometry was developed by the ancients, espe cially by the Greeks, to a high degree. But their constructions and solutions in elementary geome try were generally effected by the use only of the straight edge and compasses (instruments cor responding to the geometric elements, straight line and circle). Their achievements were, there fore, limited, and such problems as the trisection of an angle, the duplication of a cube, and all those which cannot be expressed by equations of the first or second degree, remained unsolved until the introduction of other instruments. The
word `geometry' signifies land-measure, and Herodotus attributes the origin of this science to the necessity of resurveying the Egyptian fields following each inundation of the Nile. He refers to the plan of taxation enforced by Sesostris (Rameses II.), which required a survey of the land. Proclus also confirms the Egyptian origin of geometry by saying that Thales intro duced this art from that country into Greece. The greatest among the disciples of Thales was Pythagoras, who formulated deductive geometry, and discovered many important Among the illustrious successors of Pythagoras were Anaxagoras, iEnopides, Bryson, Antiphon, Hippocrates of Chios (who duplicated the but not by elementary geometry), Zenodorus. Democritus, and Theodorus. To this list should be added the name of Plato, who introduced a new epoch in the science by formulating the method of geometric analysis, and emphasizing the necessity of accurate definition. Menwchmus, a contemporary of Plato, discovered the conic sections. Among those who studied at the Acad emy of Plato were Eudoxus, the inventor of proportion, exhaustions, and many theorems found in Euclid's Elements, and Aristotle, who improved many geometric definitions. The name of Euclid marks another epoch in the history of geometry. Euclid's work is remarkable not for its originality, but for its simplicity and per fection as a logical system, based as it was on the discoveries of his predecessors. This work of fifteen books, called the Elements, has for over two thousand years formed the basis of ele mentary instruction in geometry wherever the science has been taught. For the development of the geometry of conic sections we are indebted to Apollonius of Perga, and to Archimedes. The later Greeks also cultivated geometry enthusi astically, as is attested by Nicomedes and Hip parchus, and in the Christian Era by Ptolemy and Pappus.