An introduction to the modern theory of equations, by Florian Cajori.

202

THEORY OF EQUATIONS

It is to be noticed that, if we select every eth period in this
set, the sum of the periods thus selected is equal to one of the
known periods III, ~ 180. For instance,
'7 = 7rl +q + 1 e + + rt(,e'-)e
These periods ri', e '. 77e, **, cL e roots of ace irreducible cyclic
equation of the degree e', the coefficients of zwhich are linear fiunctions of the knozvwn periods III.
If f' is a composite number, repeat the above process by
assuming f' =e" f". If n =e e' e" *f", then the above
process calls for the solution of one equation of each of the
prime degrees e, e', e", f". As soon as one root of a cyclotomic
equation is found, the others can be obtained by raising that
one to the 2d, 3d, *.., nth powers.
183. Constructions by Ruler and Compasses. The operations
of addition, subtraction, multiplication, and division can be
performed geometrically upon two lines of given length. For
instance, in elementary geometry we learn how to construct
the quotient of a line a inches long and another line b inches
long, by the aid of the proportion x: = a: b. In elementary
geometry we learn also how to construct, by means of ruler
and compasses, the irrational -/ab. The geometric construction
of V/c + Vab is simply a more involved application of the processes just referred to. But we are not able to construct with
ruler and compasses, irrationals like /Vab. Thus it is evident
that all rational operations and those irrational operations
which involve only square roots can be constructed geometrically by the aid of the ruler and compasses.
Conversely, any geometrical construction which involves the
intersection of straight lines with each other or with circles,
or the intersection of circles with one another, is equivalent to
rational algebraic operations or the extraction of square roots.
This is the more evident, if we remember that analytically
each line and circle used in the construction is represented by