It too k nearl y tw o thousan d year s o f effor t b y mathematician s befor e a ful l
understanding o f Geometry coul d be reached, an d this required the development o f
Non-Euclidean geometry b y Gauss , Bolyai , Lobachevsk y an d th e completio n
of thei r wor k b y Klei n an d Beltrami , th e development s o n th e foundation s o f
mathematics b y Hilbert an d his definitive reformulatio n o f the axiomatic structur e
of bot h Euclidea n an d Non-Euclidea n Geometries , an d th e constructio n o f wha t
are no w calle d Riemannian Geometries b y Riemann . I n th e word s o f Herman n
Weyl, on e o f the greates t mathematician s o f this century ,
If we now look at hi s (Euclid's ) Geometry , i t i s as if we see a face whic h i s very ol d
and ver y familiar , bu t whic h i s sublimely transfigured .
In spit e o f th e fac t tha t ou r curren t vie w o f Euclidea n Geometr y a s a n ax -
iomatic disciplin e i s so much mor e subtl e tha n Euclid's , hi s work stil l compel s ou r
admiration an d veneration . H e also understoo d ver y wel l that mathematic s ha s t o
be pursue d followin g it s interna l esthetic , a s illustrate d b y th e commen t h e mad e
to hi s slave , i n respons e t o a ma n askin g hi m wha t i s th e us e o f hi s geometrica l
Give this ma n thre e penc e sinc e h e mus t need s mak e gai n b y what h e learns .
In these days when there is much clamour that onl y those parts o f science tha t
are "applicabl e t o real problems" deserv e suppor t i t will be well to heed the word s
of one of the wises t me n wh o ever lived .
Euclid's volume s als o include d man y result s o n whol e numbers , i.e. , integers .
Everyone ha s som e knowledg e o f divisibilit y propertie s o f positiv e integers . Fo r
instance 121is divisible b y 11-in fac t 121= 11x 11but 11is not divisibl e b y an y
number othe r tha n 1 and itself . Number s that posses s this last mentione d propert y
are know n a s prime numbers. Thu s
2,3,5, 7,11,13,17,19,21,23,29,31,37,41,43,47,53,59,61,67,71,73,79,
are primes , a s on e ca n easil y verify . I t i s the n natura l t o wonde r i f thi s sequenc e
keeps goin g withou t stopping , namely , whethe r th e sequenc e o f prime s i s infinite.
In Eucli d on e will find th e famou s proo f tha t thi s i s indeed so :
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