GEOMETER MOTH 



3477 



GEOMETRY 



known fossil man is the Eoanthro- 

 pus, found at Piltdown in Sussex. 

 The interpretation of historical 

 geology requires very prolonged 

 periods of time. Various esti- 

 mates based upon the rate of 

 cooling and on tidal action have 

 suggested the conclusion that geo- 

 logical time might be limited to 

 100 million years or perhaps even 

 to 20 million years. But many 

 geologists regard such estimates 

 as quite inadequate, and prefer 

 the conclusions more recently ad- 

 vanced by radio-activity, that the 

 age of the earth must be very great, 

 from 1,000,000,000 to 2,000,000,000 

 years being a reasonable estimate. 

 See Escarpment; Fault. 



Bibliography. Text- book of Palae- 

 ontology, C. A. von Zittel, Eng. 

 trans, by C. R. Eastman, 1900-2; 

 Text-book of Geology, Sir A. Geikie, 

 4th ed. 1903 ; The Natural History 

 of Igneous Rocks, Alfred Harker, 

 1909 ; The Building of the British 

 Isles, A. J. Jukes-Browne, 3rd ed. 

 1911; The Geology of To-day, J. W. 

 Gregory, 1915 ; Aids in Practical 

 Geology, G. A. J. Cole, 1919. 



Geometer Moth. Group of 

 moths whose caterpillars are often 

 called loopers from their curious 

 mode of pro- 

 gression. They 

 have only two 

 pairs of pro- 

 legs, placed 

 close to the 

 rear of the 

 body, and 

 walk by alter- 

 nately "draw- 

 ing up the 

 body into a 

 loop and then extending it again. 

 Many of these caterpillars when 

 at rest look exactly like dry 

 twigs. See Caterpillar. 



Geometric Mean. Term used 

 to denote the middle or average 

 value of tv/o quantities considered 

 from the point of view of a steady 

 rate of change from one to the 

 other. Thus the geometric mean of 

 2 and 18 is 6, for 6 is 3 times 2 and 

 18 is 3 times 6, the rate of change 

 being expressed as threefold multi- 

 plication. Expressed algebraically 

 the geometric mean of a and b is 

 *Jab. The geometric mean is more 

 correctly used than the arithmeti- 

 cal average in many investigations, 

 e.g. the mean of population at ten- 

 yearly intervals. 



Geometrical Progression. 

 Series in which the ratio, or multi- 

 plying factor, between the succes- 

 sive terms is constant. Thus in the 

 series 1, 3, 9, 27, 81 the constant 

 ratio between successive terms is 3, 

 each quantity being three times the 

 preceding one. Algebraically the 

 scries is A+Ax+Ax 2 +Ax 3 , etc., 

 or A (1+ x+x 2 +x 3 . . .). 



Geometer Moth. 



Caterpillar of Brindled 



Beauty Moth 



GEOMETRY: FROM EUCLID TO EINSTEIN 



W. D. Evans, M.A., King's College, Cambridge 



Here is given a brief historical outline of one of the oldest of 



sciences. Further information will be found under the headings 



Conic Sections ; Coordinates ; Fourth Dimension ; Mensuration, 



etc. See also Descartes ; Einstein ; Euclid 



Geometry is the science of spatial 

 relations. According to the ancient 

 belief, geometry originated in the 

 art of land-surveying, as practised 

 in Egypt, and this tradition is pre- 

 served in the Greek name (^77= the 

 earth, fierpelv = to measure.) The 

 Egyptians were certainly acquaint- 

 ed, before the year 1000 B.C., with 

 some rules of mensuration, and they 

 made practical use of the fact that 

 if the sides of a triangle are respec- 

 tively 3, 4, and 5 units, its greatest 

 angle is a right angle. 



But it was in Greek hands that 

 geometry became a logical science, 

 with general theorems. The most 

 popular text-book ever written on 

 any science was Euclid's Elements, 

 which was designed for the use of 

 students of mathematics at the 

 University of Alexandria about 

 300 B.C., and has been used as a 

 text- book of geometry in our 

 schools down to the present day. 

 Euclid begins with certain defini- 

 tions, axioms and postulates, from 

 which he professes to deduce all his 

 results by purely logical processes, 

 without further appeal to the eye 

 or to common-sense. Thus he 

 thinks it necessary to prove that 

 two sides of a triangle are together 

 greater than the third, an example 

 which illustrates the abstract philo- 

 sophical outlook of the Greek geom- 

 eters. Though modem scrutiny has 

 detected some flaws in Euclid's 

 logic, and many of his methods 

 have been abandoned as cumber- 

 some, his deep insight into some of 

 the most difficult problems of 

 geometry is undoubted, and has 

 been attested by some of the best 

 modern writers. 



Euclidean Theorems 



Euclid's propositions are of two 

 kinds, theorems and problems ; a 

 theorem establishes a geometrical 

 property by deduction from previ- 

 ous results ; a problem is a method 

 of making a geometrical construc- 

 tion, followed by a theoretical proof 

 that the method leads to the result 

 desired. Beginning with proposi- 

 tions concerning simple figures 

 bounded by straight lines, such as 

 triangles, squares, rectangles, and 

 parallelograms, Euclid passes to 

 the geometry of the circle and of 

 regular polygons with more than 

 four sides. After a preliminary 

 study of ratio and proportion, the 

 properties of similar figures (of like 

 shape but of different dimensions) 

 are discussed, similar triangles 

 being the leading case. 



This work occupies the first six 

 books of the Elements, which are 

 devoted to the geometry of figures 

 in one plane (plane geometry) ; 

 four books follow on arithmetic, 

 and then the eleventh, twelfth, and 

 thirteenth books consider the geom- 

 etry of figures in three-dimensional 

 space, and of solid bodies (solid 

 geometry). The only curved line 

 discussed by Euclid was the circle, 

 but the Greeks also studied in great 

 detail the geometry of the conic 

 sections, which include three types 

 of curves, the parabola, ellipse, and 

 hyperbola. These curves are the in- 

 tersections of the ordinary (right 

 circular) cone by different "planes ; 

 they may be illustrated by the 

 shadows of the circular base of a 

 candlestick cast on the floor and 

 walls of a room by the candle, held 

 in different positions. 



Applied Geometry 



The classical geometry of the 

 Greeks was not, strictly speaking, 

 numerical, though it involved ratio 

 and proportion, but their astro- 

 nomy, which they developed to a 

 considerable degree of accuracy, 

 demanded a method of measuring 

 angles. The division of the right 

 angle into 90 equal parts, called 

 degrees, and of the degree into 60 

 minutes, they derived from the 

 Babylonians ; the measurement of 

 certain lengths connected with an 

 angle of given size was a natural 

 step forward, and this led to the 

 science of trigonometry. These 

 lengths, or, as they may be more 

 accurately described, ratios of 

 lengths, have received the name of 

 trigonometrical ratios or functions 

 (sine, cosine, etc.), and plane trigo- 

 nometry deals with their application 

 to the measurement of triangles. 



This science is applied to the 

 surveying and mapping of small 

 areas of the earth's surface ; when 

 the areas are so large that the 

 spherical shape of the earth must 

 be considered, spherical trigonom- 

 etry is required. This is the study 

 of triangles on a sphere, bounded 

 by arcs of great circles that is, 

 such circles as divide the surface of 

 the sphere into two equal parts 

 and is essential to astronomy and 

 navigation. Astronomy in later 

 times utilised the properties of the 

 conic sections, for Kepler hi A.D. 

 1609 found that the planets moved 

 round the sun in ellipses, and the 

 known properties of the ellipse led 

 Newton to the discovery of the law 

 of gravitation. 



