The Indianapolis times. [volume] (Indianapolis [Ind.]) 1922-1965, 1933-08-04
-Dietz on Science
'IMP0SSIBLE'IS
APPLICABLE TO
VARIOUS POINTS
Squaring the Circle and Tri-
Sectlng an Angle Can't
Be Done.
BY DAVID DIETZ.
Seripps- Howard Science Editor

MuSsouni. IN a recent interview
with Emil Ludwig is quoted as say
ing. There Is no such thing as
the impossible.
Dr. Edward Kasner. professor OF
mathematics at Columbia univer~
sitv comments on that by saying 1t
NAV be true in khe domain Of
craftsmanship or physical invention
but that il certainly IS not true IN
mathematics
Three problems upon which many
men have spent long hours OF
thought and work through the CEN
turtles are all impossible, Kasner
says. They are "Squsring the circle,
doubling the cube, and "trisect-
ing an angle
Ever since the days Of the Greek
geometers. savants and their anna
teur gounterparts have tried to
solve these problems.
By "squaring the circle Is meant
the construction Of A square equal
in area 30 given circle, by the use
OF only two instruments, namely the
ruler and the compass. It is also
specified that the instruments must
he used a finite number of times.
Obviously. any process requiring an
infinite number OF steps is no real
solution.
II is. Of course, easy enough to
construct a square equal in area to
a circle If we may use other meas
uring instruments But the problem
is 10 do It with compass and ruler
by the axioms of Euclid's geometry
Pr Enters Picture
The Greeks couldn't solve the
problem, neither could the Romans,
nor the Arabic and Jewish savantS
who kept alive the torch of learn
ing after the fall of the Roman en-
pire European scholars attacked the
problem without avail in the Re-
naissance and thereafter.
Now DR. Kasner points out that
just because a problem seems diff
cult is no sign 1t may not have a
solution. For example, geometers
tried for centuries TO construct
regular polygon Of seventeen sides.
But in 1876. Gauss. then 10 years
old. found an elementary conStruc-
ton
The case of squaring the circle,
however. is different. In 1852, Lin
demann. a German mathematician,
showed that the problem was i'm
possible. His proof is based upon a
demonstration of the fact hat "pi.
the ratio between the circumference
and diameter of circle, is trans
cendental
BY "transcendental" the mathe
maticians mean that it is incapable
of being defined by any combina-
ton Of a finite number of co-effic-
tents" Maybe YOU had just better
take their word for that.
The point is that with ruler and
compass. It is only possible TO per-
form what are known as rational
operations and the extraction of
square roots. Since squaring the
circle involves ''pi," and since this
number can not be expressed by
such operations, the problem can
not be solved.
Futility Known in 1882
DR.. Kasner says that since 1882
no one has the moral right to waste
his time trying TO square the circle.
The doubling of the cube" that
is. the construction of a cube OF
double the volume of a given cube,
is the second problem which is inn
possible. DR.. Kasner tells us. That
is because it involves the cube root
of two and there is no way OF OBJ
tanning it with ruler and compass.
The problem of doubling square.
however. is a very easy one. You
probably knew the answer when
YOU were a sophomore at high school
although YOU may have forgotten
II by now.
Merely draw the diagonal of the
square and erect square upon IT..
This new square has double the area
of the original one.
SEE now if YOU remember enough
algebra and geometry TO demon
strate the proof.
II is impossible 10 trisect an angle
with only compass and ruler be-
cause this problem also Involves
cube roots which can not be con-
structed with those two instruments
alone.
Another impossible"
Some readers may object to the
use OF the word impossible in any
statement But DR. KaSner points
out that il is not difficult TO set up
impossible examples in mathematics.
For example. try 10 find TWO odd
numbers whose sum is 81
Ii looks very simple, doesn't It?
But it is impossible just the same.
The reason? The sum of any two
odd numbers is always an even
number. Try few examples and
see for yourself.
There are plenty Of other i'm
possible problems known 10 maihe-
mates. although amateur mathe-
maticians seem TO concentrate on
the three already mentioned.
TAHE NTANE