156 



GEOMANCY 



GEOMETRY 



the standard text-books of geology ; see also article 

 STRATUM. For Experimental Geology, see Daubree's 

 Etudes Synthetiques de Geologic Experimental (1879). 

 For works dealing with Palaeontology, see under that 

 article. For Physiographical Geology, see Memoirs of 

 Geological Surveys of British Islands, passim; -Ramsay's 

 Physical Geography and Geology of Great Britain ( 1878 ) ; 

 A. Geikie's Scenery and Geology of Scotland (1889); 

 Hull's Physical Geography and Geology of Ireland ( 1878 ) ; 

 Dutton's ' Tertiary History of the Grand Canon District,' 

 Monographs of U.S. Geol. Survey (vol. ii. 1882); also 

 Annual Reports of U.S. Geol. and Geograph. Survey of 

 Territories (1867-78), passim; De la Noe and De 

 Mar 'erie, Les Formes du Terrain (1888). For Geo- 

 logy of British Islands, see Maps and Memoirs of the 

 Geological Survey ; works by Ramsay, A. Geikie, and Hull 

 already cited ; Woodward's Geology of England and Wales 

 ( 1887 ) ; Kinahan's Geology of Ireland ( 1878 ) ; Murchison's 

 Siluria ( 1867 ) ; Macculloch's Western Islands of Scotland 

 (1819); Nicol's Guide to the Geology of Scotland (1844) 

 these last two works rather out of date ; Miller's Old 

 Red Sandstone (1858); Green. Miall, and others, Coal: 

 its History and Uses ( 1878 ) ; Hull's Coalfields of Great 

 Britain ( 1881 ) ; Meade's Coal and Iron Industries of the 

 United Kingdom ( 1882 ) ; Phillips' Geology of Oxford 

 and the Valley of the Thames (1871), and Geology of 

 the Yorkshire Coast (1875); Tate and Blake, The York- 

 shire Lias (1876). For further references to treatises 

 dealing with the geology of England and Wales, see especi- 

 ally Woodward's work cited above. The following works 

 deal witli Pleistocene Geology and the Antiquity of Man : 

 Lyell's Antiquity of Man (1873); Lubbock's Prehistoric 

 Times (1878) ; Evans' Ancient Stone Implements of Great 

 Britain (1872); Dawkins' Cave-hunting (1874), and 

 Early Man in Britain (1880) ; J. Geikie's Great Ice Age 

 (1877), and Prehistoric Europe (1881); Dawson, The 

 Earth and Man ( 1887 ) ; De Quatref ages, The Human 

 Species (1879) ; Joly's Man before Metals ( 1883) ; Penck's 

 Die Vergletscherung der deutschen Alpen (1882) ; Falsan, 

 La Ptriode Glaciaire ( 1889) ; Wright's Ice Age in North 

 America, <Scc. (1889). For treatises bearing on Geological 

 Climate, see Croll's works already cited; also J. D. 

 Whitney, The Climatic Changes of Later Geological 

 Times ( 1882). Amongst works on Economic Geology the 

 following may be mentioned : Page's Economic Geology 

 (1874); Williams' Applied Geology (1886); 'Penning's 

 Engineering Geology (1880); Nivoit's Geologic applique'e 

 a I' Art de I'lm/rnieur (1887). For methods of geological 

 observation and the making of geological maps, see the 

 larger text-books, Sir A. Geikie's Outlines of Field Geology 

 (1879), and Penning's Field Geology (1876). Sir A. 

 Geikie, The Founders of Geology (1897), deals with 

 Desmarest, Gviettard, and other early geologists. 



Geomancy. See DIVINATION. 



Geometrical Mean of two numbers is that 

 number the square of which is equal to the product 

 of the two numbers ; thus, the geometrical mean of 

 9 and 16 is 12, for 9 x 16 = 144 = 12-. Hence the 

 geometrical mean of two numbers is found by 

 multiplying the two numbers together, and extract- 

 ing the square root of the product. 



Geometrical Progression. A series of 

 quantities is said to be in geometrical progression 

 Avhen the ratio of each term to the preceding is the 

 same for all the terms i.e. when any term is equal 

 to the product of the preceding term and a factor 

 which is the same throughout the series. This 

 constant ratio or factor is termed the common ratio. 

 For example, the numbers 2, 4, 8, 16, &c., and 

 also the terms a, ar, ar 2 , ar\ &c., are both ex- 

 amples of geometrical progression or series. The 

 sum of such a series is obtained as follows : Let a 

 be the first term, n the number of the terms whose 

 sum, s, is required, and let r be the common ratio. 

 Then s = a + ar + ar 2 + . . . + ar n ~ ' ; also from 

 multiplication of both sides of this equation by 

 r, sr = ar + ar 2 + ar* + . . . + ar". Subtraction 

 of the former from the latter expression gives sr - 

 s = ar n -a; or s(r - 1 ) = a(r n - 1 ), and hence 



Geometry is that branch of the science oi 

 mathematics which treats of the properties of space. 

 When the properties investigated relate to figures 

 described or supposed to be described on space of 

 two dimensions, there arise such subdivisions as 

 plane and spherical geometry, according to the 

 surface on which the figures are drawn. If the 

 properties relate to figures in space of three dimen- 

 sions they fall under what is called solid geometry, 

 or now more frequently, geometry of three dimen- 

 sions. Again, from the mode in which the pro- 

 perties of figured space are investigated, arise two 

 other subdivisions, pure and analytical geometry. 

 The somewhat arbitrary subdivision into element- 

 ary and higher geometry arises from the fact that 

 the geometrical books of Euclid's celebrated work, 

 the Elements, treated only of plane figures com- 

 posed of straight lines and circles, of solid figures 

 with plane faces, and of the three round bodies, the 

 sphere, the cylinder, and the cone. 



Other subdivisions of geometry arise from the 

 threefold classification that may be made of the 

 properties of space. These properties may be topo- 

 logical, graphical, metrical. The first class of pro- 

 perties are independent of the magnitude or the 

 form of the elements of a figure, and depend only 

 on the relative situation of these elements. Per- 

 haps the simplest example that could be given of 

 this class of properties is that if two closed contours 

 of any size or shape traverse one another, they 

 must do so an even number of times. No systematic 

 treatise on this part of geometry has ever been 

 drawn up, and it is only in papers scattered here 

 and there in scientific journals that contributions 

 towards such a treatise are to be found. The prin- 

 cipal names under which such contributions are to 

 be looked for are Euler, Gauss, Listing, Kirkman, 

 and Tait. 



The graphical or protective properties of space, 

 which constitute the subject of projective geometry, 

 are those which have no reference to measurement, 

 and which imply only the notions of a straight line 

 and a plane. A simple example of this class of 

 properties is the well-known theorem of Desargues : 

 If two triangles be situated so that the straight 

 lines joining corresponding vertices are concurrent, 

 the points of intersection of corresponding sides are 

 collinear, and conversely. 



The metrical properties of space are those which 

 are concerned with measurement. An example of 

 a metrical property is the theorem of the three 

 squares : The square on the hypotenuse of a right- 

 angled triangle is equal to the sum of the squares 

 on the two sides. The geometry of Euclid's Ele- 

 ments is metrical. 



Descriptive geometry is not so much a part of 

 science as an art. It has for its object to represent 

 on a plane which possesses only two dimensions, 

 length and breadth, the form and position in space 

 of bodies which have three dimensions, length, 

 breadth, and height. This object is attained by 

 the method of projections. 



Analytical geometry is a method of representing 

 curves and curved 

 surfaces by means 

 of equations. Be- 

 fore showing, how- 

 ever, how a curve 

 can be represented 

 by an equation, it 

 will be necessary 



to explain what is x 



meant by the co- 

 ordinates of a point. 

 If two axes, XX', 

 YY', cutting each 

 other perpendicularly be taken, the position of a 

 point P in the same plane as the axes is determined, 



Y/ 



