The Probert Encyclopaedia of Science & Technology

GEOMETRY

Geometry (from the Greek ge, earth, and metron, measure), as its name implies, was primarily the mathematical science which had for its object the measurement of portions of the earth's surface; but now geometry may be termed the science which treats of the properties and relations of definite portions of space, such as surfaces, volumes, angles, lines. The relation between the parts of the same figure may be of two kinds, of position or of magnitude; for example, two points in a straight line, four points on the same circle, two straight lines perpendicular to one another, a straight line tangent to a circle, are relations of position. On the other hand, the proportionality of homologous lines of two similar figures, the equality of the square constructed on the hypotenuse of a right-angled triangle to the sum of the squares constructed on the sides containing the right angle, that of the volumes of two pyramids on equal bases and of the same height, are relations of dimension. But the relations of position govern the relations of dimension, and vice versa; that is, the one set of relations depend upon the other. Thus it ia because a triangle is rectangular that the square constructed on one of its sides ia equivalent to the sum of the squares constructed on the other two, and, vice versa, that relation between the magnitudes of the squares on the three sides depends on the triangle being right-angled. The geometer may draw indifferently from the study of a figure either the knowledge of the relations of position or that of the relations of dimension, on the condition that he knows how to apply relations of the one kind to those of the other: and the principal aim of geometry is to examine into the connection between the relations of magnitude and those of position.

Geometry may be conveniently divided into several principal sections - elementary geometry, practical geometry, analytical geometry, infinitesimal geometry, etc.

Elementary geometry consists of two parts - plane geometry, the object of which is the study of the simplest figures formed on a plane by straight lines and circles; and solid geometry or geometry of three dimensions, which treats of straight lines and planes considered in any relative position whatever, of figures terminated by planes, of the cylinder, of the cone, and of the sphere.

Analytical geometry, either plane or solid, makes use of the method of co-ordinates introduced by Descartes and primarily applied to curves. In ancient times, though curves were studied and the principal properties of conic sections known, still no connection existed between these curves, nor was there any means of establishing one, so that the study of one was of no value to that of another. The first question in introducing the analytic method was then to fix upon some means which should serve to construct every curve by successive points as numerous and as closely brought together as is necessary in order to lay down the curve, Now the position of a point in a plane may be determined by two intersecting perpendiculars drawn from two fixed lines - the co-ordinate axes - at right angles to each other. An equation may then be found which states the relation between the co-ordinates of any point, that is, its distance from the two co-ordinate axes. The study of the curves will thus be simply the study of their equations. In this way a typical equation for a curve in a certain system may be got, so that if at another time the curve is represented under another definition in investigating its equation in the same system of co-ordinates, particularized so aa to simplify as much as possible the calculations, it will suffice to compare the particular equation with the general one to verify the identity of the curve, to give it its name, and to know all the properties of it which have been studied previously. In a similar way the analytical geometry of solid bodies is based on the fact that the position of any point in space can be determined by reference to three intersecting planes.

Infinitesimal geometry is simply a continuation of the analytical geometry of Descartes, of which it may indeed be said it forms a part; the difference consists simply in the nature of the questions which, as they involve the measurement of magnitudes, the incessantly variable elements of which cannot be summed up by finite parts, require the use of the infinitesimal calculus.

Descriptive geometry consists in the application of geometrical rules to the representation of the figures and the various relations of the forms of bodies according to certain conventional methods. In the descriptive geometry the situation of points in space is represented by their orthographical projections, on two planes at right angles to each other called the planes of projection.
Research Geometry