Evening star. [volume] (Washington, D.C.) 1854-1972, 1908-09-27

COME interesting fallacies may be found in
~ mathematics One of the Oldest is the
following. db is equal to square. Therefore.
ab- b. equals d2-b2 Therefore, (d-b1 equals
"a b. ia b.. Therefore, b equals a b.
Therefore, b equals 2b Therefore, equals ,.
Another similar to it may be written: e'
equals I. Square both sides, and ez' equals
l. Therefore. ~ equals o. Therefore, equals
o. Therefore, CO equals GO But ca equals Il
and G. equals l. Therefore, i equals
The following problem appeared IN Pans IN
1901 If the product OF two numbers is equal
to two other numbers, the numbers Will be IN
proportion, and if in proportion the hrst term
is greater than the second term the third term
is greater than the fourth. Thus, II dd is equal
to bt we may obtain the proportion q b :: C. d.
and then If a is greater than b. c Is also greater
than d The equation IS satisfied II we take
values as follows: a equals d equals b equals
{ equals l. This shows that the value of a is
greater than the value of b, since I IS greater
than i. But since is greater than d, Il must
be greater than l.
The following fallacy was printed in one of
the magazines about fifteen years ago: Sup
pose three coins are thrown up, and We wish to
determine the probability OF all three coming
down alike. Since each must come with either
head or tail uppermost, either head or tail has
exactly one chance IN two OF beng upper
most From this it follows that the chance of
all three appearing the same is equal to the
cube of II, or 1IS. This is true for either
heads or tails. Hence the probability of
all appearing either head tail IS "a.
But at least two must come up alike, since
there are only two sides, and of the third coin
the probability is I 2 that it comes up head,
l. ? that it comes up tail Thus, the probability
that it comes down so as to show the same side
as the other two is ?, and therefore the prob
ability that it will come down the same the
other two is 1j2 Therefore, the probability
that all three will come down alike is Il
Euclid prepared long list of geometrlc fall
laCles, which has been lost There are, how
ever many known geometric fallacies that test
the ingenuity of the worker. Among these
NAV be mentioned such fallacies as to prove
that right angle is greater than right angle:
to prove that part of line is equal to the
whole line; to prove every triangle isosceles;
that if two opposite sides of quadrilateral are
prove that pi' is equal to Sl3 or 2i3,
when everyone knows that it is equal APP
proximately to 3.I4Io, or about 3 III.
curious paradox was propounded by Zeno
many centuries ago, He said it IS evident that
an arrow may not move where it is not: and
since it is Just as impossible for it to move
where it is. / {. in space that it completely
fills, 1t follows that arrow cannot move at
all; and therefore all motion is impossible
To the same is due the famous problem
about Achilles and the tortoise. He claimed
that if Achilles ten times fast as the
tortoise, yet if the tortoise had 1000 yards
start it could never be overtaken; for when
Achilles had gone 1000 yards the tortoise
Achilles had gone the yards the tortoise
would be yards ahead, and so on forever.
Achilles would thus get nearer and nearer to
the tortoise; but would never be able to over
take him. Zeno made the divisions of time