GEOMETRY 



2447 



GEOMETRY 



Proof. Draw AD perpendicular to BC. 

 Then AB2 BCxBD 



and AC2 = BCXDC (for it has been 

 proved that "if in a right triangle an altitude is 

 drawn to the hypotenuse, each leg of the triangle 

 is a mean 'proportional between the hypotenuse 

 and the adjacent segment"). 



Adding, AB~2 + AC2 = BC (BD + DC) or =BC2 

 (for by axiom, "if equals be added to equals, the 

 results are equals"). Q.E.D. 



Another theorem which lies at the basis of 

 many propositions in geometry, and whic,h is 

 also credited to Pythagoras, is the one which 

 declares that: 



The sum of the three angles of a triangle is 

 equal to two right angles. 



Hypothesis: ABC is any triangle. 



Conclusion: Angle A + angle B + angle C = 

 two right angles. 



Proof. Draw 

 MN through B \ 



parallel to AC, 

 and produce AB 

 and CB, forming 

 angles a, b and c. 



Then angle A= 

 angle a, being 

 corresponding an- 

 gles of parallel 

 lines, which have 

 been proven equal ; A 

 and B = angle b, 

 being vertical angles, which have been proven 

 equal ; 



and angle C = angle c, corresponding angles of 

 parallel lines. 



Adding, angle A + angle B + angle C = angle a + 

 angle b + angle c (for by axiom, "if equals be 

 added to equals the results are equal" ) . 



But angle a + angle b + angle c = two right an- 

 gles, for "the sum of all the angles that can be 

 found at a point in a straight line and on the 

 same side of the line, is equal to two right angles." 



It is impossible to list here all the theorems 

 which are especially interesting; indeed, the 

 interest is not fully apparent when they are 

 inspected in this disjointed manner. Only as 

 the student follows logically from one theorem 

 to the next can he feel the full joy that geom- 

 etry has to give. 



Why Geometry Is Studied in School. Geom- 

 etry is usually accepted without question as 

 one of the important subjects in a secondary 

 school course. Few arguments have to be 

 made for it, so apparent is its value. First of 

 all, it lies at the basis of the whole science of 

 measurement. To be sure, now that the theo- 

 rems and problems have been fully worked 

 out, the results have been in a measure passed 

 on to arithmetic and are there stated as rules, 

 so that a person may figure out the amount 

 of carpet needed for a certain floor, the cubic 



capacity of a wheat bin or of a hogshead, the 

 height of a tree, or many of the other practi- 

 cal problems without knowing the fundamen- 

 tal principles of geometry on which these 

 problems are based. However, knowledge of 

 geometry enables him to apply his rules far 

 more intelligently, as well as to work out new 

 applications for himself. 



But geometry has other values, as important 

 if not so practical. Its logical plan, the way it 

 proceeds from step to step without allowing 

 for any gap in the reasoning, develops the 

 reasoning powers in a way impossible to any 

 subject which admits of more hit-or-miss 

 methods. Nor are all the theorems and prob- 

 lems of geometry worked out and set down in 

 the textbooks for the student to memorize. 

 Any good book contains a large number of 

 original propositions which the student must 

 work out for himself, and nothing proves more 

 clearly his mastery, not only of geometric prin- 

 ciples but of the workings of his own mind, 

 than the ability to work out a theorem clearly 

 and with the fewest possible statements make 

 his demonstration clear to others. There is a 

 joy which must be felt to be appreciated in 

 detecting a fallacy or a gap in the reasoning, 

 or in tracing out the definition, axiom or 

 proposition which makes it plain that some 

 elusive point is actually provable. 



To sum up, no study can quite take the place 

 of geometry as a cure for slovenly habits of 

 thinking, and until some such substitute can 

 be found it seems likely to hold its place, even 

 for those who have no practical use for its 

 teachings. E.D.F. 



Consult Klein's Famous Problems in Elemen- 

 tary Geometry; Smith's "History of Modern 

 Mathematics" in Merriman and Woodward's 

 Mathematics. First books in geometry may be 

 purchased from any schoolbook publishing house. 



Related Subject*. In addition to the topics 

 referred to in the above discussion, there are in 

 these volumes many articles which are more or 

 less closely connected with geometry. The reader 

 is referred to the following: 



Angle 



Area 



Axiom 



Circle 



Cone 



Cube 



Curve 



Cylinder 



Degree 



Ellipse 



Line 



Magnitude 



Mensuration 



Plane 



Polygon 



Prism 



Pyramid 



Quadrilateral 



Rectangle 



Rhomboid 



Rhombus 



Sphere 



Square 



Square Measure 



Trapezium 



Triangle 



