GEOMETRY

(Yunanca köken — orijinale bakınız.) The science which treats of the properties of figured space. The etymology of the term suggests the object to which geometry was first applied, viz., the measurement of land. It is pretended that the science was invented in Egypt, where the annual overflowing of the waters of the Nile obliterated the land marks, and rendered It m to have recourse to measurement in order to ascertain the proper allotment of each individual; but whatever mav have been the origin of the term, the occasions on whltll it is nece ary to compare things with one another in respect of their forms and magnitudes are so numerous in everv stage of society, that a geometry more or le perfect must have existed since the first dawn of civilization. Objects of Geometry. — In geometry bodies are considered only in reference to the properties of extension or niagni rude, figure, and divisibility. Every body occupies in hide finite space a certain determinate place, or finite portion of space, which is called its volume. The limits or boundaries which distinguish the place of the body, and separate it from the surrounding space, are called surfaces; a surface is, therefore, common to the two portions of space which it separates. As the limitation of space gives rise to the idea of surface, so the limitation of surface produces Unet, a line being the boundary of n surface, or the place In which two surfaces intersect each other, and therefore common to both. In like manner, the limitation of a line, or the intersection of two lines, produces a point. But a point marks only position, and has no properties. A line has length; ■ surface length and breadth; and a volume length, breadth, and thickne . Hence the properties of lines, the proper tics of surfaces, and the properties of volumes or solids, comprehend the objects of geometry. Although the notion of a point is acquired from the cou sideration of lines, that of a line from the consideration of surfaces, and that of a surface from the consideration of bodies or material objects, it does not follow from this that points, lines, and surfaces are themselves really material. Geometry regards all bodies in a state of abstracts.n very different from that in which they actually exist; and the truths which it discovers and demonstrates are pure al>- str actions — hypothetical truths, which are not, however, on that account, the le useful. For example, it is impo ible byany mechanical means to draw a line absolutely straight, or to describt. a perfect circle; but the nearer the Hue approaches to perfect straightne , and the more accurately the circle is described, the nearer will their properties np proach to those of the ideal straight lines and circles which are the objects of geometrical consideration. The theorems of geometry are, therefore, not strictly true in their npplica tion to material bodies, but they approximate sufficiently to truth for all practical purposes. They enable us to ascer tain, with all the precision of which our senses are capable, the distances of inacce ible objects, the dimensions of a given surface, the contents of a given solid; to compute tha distances and motions of the planets; to predict the celestial phenomena; and to navigate a ship from any given point of the globe to any other. Divisions of Geometry. — Geometry is divided into elementary and transcendental. Elementary geometry treats only of the straight line and circle; of figures bounded by straight lines and circles; and of solids bounded by these figures. The circle is only the curve line introduced into the elements of geometry; the simplicity of its description, the ease with which many of its most useful properties nre deduced, and the nece ity of making use of it in the simplest constructions — such as raising a perpendicular, measuring an angle, and even making one straight line equnl to another—being reasons for this preference. The construction of algebraic equations of the second degree, and in general all problems that can be solved by means of straight lines and circles, are also referred to elementary geometry. Transcendental geometry, properly speaking, is that which has for its object all curves different from the circle; as the conic sections, and curves of the third and higher orders. It comprehends, also, the construction of equations of the third and fourth, as well as of the higher degrees. But some writers understand by transcendental gejmctry the applications of the differential and Integral calculus to the investigation of the properties of curve lines and surfaces. Geometry is also divided Into ancient and modern; ancient geometry being that form of demonstration and investigation which was employed by the Greeks, and of which Euclid's E'ements form a well-known example; modern geometry that in which algebra, or the differential or integral calculus. Is made use of. We also speak of pure geometry, practical geometry, and applied geometry. Descriptive geometry has already been considered under that term. 513 [s. 527]