It i s widel y agree d tha t Car l Friedric h Gauss' s 1801boo k Disquisi-
tiones Arithmeticae [G ] was the beginnin g o f modern numbe r theory ,
the firs t wor k o n th e subjec t tha t wa s systemati c an d comprehen -
sive rather tha n a collectio n o f special problem s an d techniques . Th e
name "numbe r theory " b y whic h th e subjec t i s know n toda y wa s i n
use a t th e time—Gaus s himsel f use d i t (theoria numerorum) i n Arti -
cle 5 6 o f th e book—bu t h e chos e t o cal l i t "arithmetic " i n hi s title .
He explaine d i n th e first paragrap h o f hi s Prefac e tha t h e di d no t
mean arithmeti c i n th e sens e o f everyda y computation s wit h whol e
numbers bu t a "highe r arithmetic " tha t comprise d "genera l studie s
of specifi c relation s amon g whol e numbers. "
I too prefer "arithmetic " t o "numbe r theory." T o me, number the -
ory sounds passive , theoretical , an d disconnecte d fro m reality . Highe r
arithmetic sound s active , challenging , an d relate d t o everyda y realit y
while aspirin g t o transcen d it .
Although Gauss' s explanatio n o f what h e means by "highe r arith -
metic" i n his Prefac e i s unclear, a strong indicatio n o f what h e had i n
mind come s a t th e en d o f hi s Prefac e whe n h e mention s th e materia l
in hi s Sectio n 7 o n th e constructio n o f regula r polygons . (I n mod -
ern terms , Sectio n 7 i s th e Galoi s theor y o f th e algebrai c equatio n
xn
1 = 0. ) H e admit s tha t thi s materia l doe s no t trul y belon g t o
arithmetic bu t tha t "it s principle s mus t b e draw n fro m arithmetic. "
IX
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