GEOMETRY 



3478 



GEOMETRY 



A great advance in geometry was 

 made in 1637 by Rene Descartes, 

 who introduced the method of co- 

 ordinates, which lies at the base of 

 analytical geometry. The idea of 

 coordinates is simplicity itself ; it 

 is that the position of any point in 

 a plane may be represented by its 

 perpendicular distances from two 

 fixed perpendicular lines. For ex- 



ample, in the figure, X' O X and 

 Y' Y are the fixed axes, P L is 

 equal to 3 units, and P K to 2 units, 

 and P is represented by its co- 

 ordinates (3, 2). To distinguish P 

 from Q, R, S, which are at the same 

 distances from the axes as P, nega- 

 tive coordinates are used. Thus Q 

 is the point ( 3, 2), R is ( 3, 2), 

 S is (3, 2). In genera], the per- 

 pendicular distance of a point from 

 Y is denoted by the letter x, and 

 the distance from X by the letter 

 y, and the point is called (x, ?y). 

 Value of Coordinates 



This simple notation had far- 

 reaching effects in geometry, and 

 has enabled geometrical concepts 

 to be applied with great advantage 

 to other branches of mathematics, 

 such as the differential and integral 

 calculus, mechanics and electricity. 

 Its utility depends on the fact that 

 a curve may be regarded as an 

 assemblage of points possessing a 

 certain common property ; e.g. a 

 circle is an assemblage of points all 

 at the same distance from the fixed . 

 centre ; when considered in this 

 way a curve is called a "locus." 



This common property may be 

 expressed in the form of an alge- 

 braical equation connecting the x 

 and the y of any and every point 

 on the locus, and the curve is then 

 completely represented by this 

 equation, which implicitly contains 

 every possible property of the 

 curve. Thus the conic sections can 

 all be represented by an equation 

 of the form 



OX--+ 2hxy+by-+ 2gx+2fy+c=o, 

 which for different numerical 

 values of the constants a, h, b, g, f, c 

 may denote a circle, a parabola, 

 an ellipse, or a hyperbola ; and the 

 properties of these curves can all 

 be deduced from this equation. By 

 the discovery of analytical geo- 

 metry the scope and generality of 

 geometrical methods was immense- 

 ly increased, and an even greater 



degree of success attended the 

 application of the method of co- 

 ordinates to three-dimensional geo- 

 metry. In analytical solid geo- 

 metry three coordinate planes 

 (such as the floor and two adjacent 

 walls of a room) take the place of 

 the coordinate axes, and the posi- 

 tion of a point is represented by 

 three coordinates (x, y, z). ' An 

 equation between x, y, and z then 

 denotes a surface. 



Practically all advances in solid 

 geometry have been due to analy- 

 tical methods, on account of the 

 impossibility or difficulty of repre- 

 senting solids in a plane. For this 

 reason little advance in solid geo- 

 metry was made by the ancients. 



Line geometry is the name given 

 to that system of geometry in 

 which straight lines replace points 

 and systems of straight lines 

 systems of points. H. Grassman 

 (1844) and Cayley (1859) and J. 

 Pliicker were the three chief ex- 

 ponents of the system, which in 

 new hands and those of modern 

 geometers has added greatly to the 

 knowledge of the properties of 

 surfaces and solids. 



Another great advance which 

 may be compared in generalising 

 power to that made by Descartes, 

 though its effects have not been so 

 far-reaching, was the introduction 

 of projective geometry, the founda- 

 tions of which were laid about the 

 same time by Desargues. The 

 germ of projective geometry is 

 already implicitly contained in the 

 idea of the sections of a cone, which 

 may be circles, ellipses, parabolas, 

 or hyperbolas. Straight lines 

 drawn from the vertex of the cone 

 to meet the circular base, itself a 

 section of the cone, will also meet 

 any other section of the cone, an 

 ellipse, for example, and two such 

 curves as this circle and this &\ipse 

 may be called projective. 



Orthogonal Projection 



Certain properties are common 

 to curves which are projective, and 

 by utilising these properties a con- 

 nexion is obtained between theo- 

 rems which are true for the differ- 

 ent kinds of conic sections. In 

 particular, properties of the other 

 conic sections can be inferred from 

 known properties of the circle. 

 Another kind of projection of great 

 usefulness is orthogonal projection, 

 in which a curve is projected from 

 one plane on to another by means 

 of straight lines perpendicular to 

 the second plane. For example, a 

 section of an ordinary (right circu- 

 lar) cylinder by a plane not parallel 

 to the base is an ellipse ; the cir- 

 cular base may be considered as the 

 orthogonal projection of the ellipse. 

 This method is the basis of practi- 

 cal solid geometry, which enables 



us to represent three-dimensional 

 objects accurately on a plane. 



The axioms and postulates on 

 which Euclid based his system of 

 geometry are accepted with slight 

 modification as the foundation of 

 trigonometry, analytical geometry, 

 and projective geometry, and the 

 successful applications" of these 

 sciences in practice bear witness to 

 the substantial truth of these 

 axioms. But geometry may be 

 considered from a purely abstract 

 standpoint, as a science in which 

 certain theorems regarding points, 

 lines, planes, etc., are logically 

 deduced from certain premises, 

 with no necessary connexion with 

 the space of experience, and it has 

 been possible to construct per- 

 fectly consistent theories on the 

 basis of a denial of some of Euclid's 

 assumptions. 



Non-Euclidean Geometry 



Many perfectly logical non- 

 Euclidean systems of geometry 

 have been evolved, the two chief 

 of which are known as elliptic and 

 hyperbolic geometries. These geo- 

 metries are leading to new con- 

 cepts of space. 



For instance, the parallel postu- 

 late of Euclid amounts to the 

 assertion that through a given 

 point only one straight line can be 

 drawn parallel to a given straight 

 line ; if we assume that two paral- 

 lels or no parallel can be drawn we 

 are led to different kinds of non- 

 Euclidean geometry, each perfectly 

 consistent with itself, though lead- 

 ing to conclusions apparently in- 

 consistent with experience. But it 

 is conceivable that space may be 

 really non - Euclidean, although 

 apparently Euclidean in such com- 

 paratively small parts as we are able 

 to explore, just as a sheet of water 

 appears plane, though we know it 

 is really part of the curved surface 

 of the earth. This possibility has 

 recently received strong support 

 from the researches of Einstein. 



Among more recent develop- 

 ments of geometry we may men- 

 tion the theory of vectors ; a vector 

 is essentially a straight line given 

 in magnitude, direction in space, 

 and direction along its length (a 

 straight line with an arrow-head on 

 it, in fact), but not fixed in position. 

 This theory has had many interest- 

 ing physical applications, and of 

 late especially in connexion with 

 four-dimensional space a purely 

 mathematical conception in which 

 the passage from three to four di- 

 mensions is imagined as analogous 

 to the passage from two dimensions 

 to three. This conception seems 

 sufficiently remote from experience, 

 yet it has played an important 

 part in the development of the re- 

 cent physical theory of "relativity." 



