The progressive farmer. [volume] (Winston, N.C.) 1886-1904, 1891-04-28

EgypG 5UO years before the Exodus
OF the Israelites. Among the Greeks,
the problem never ceased to play a
part that greatly influenced the prog
ess of mathematics. In the middle
ages, also, the squaring of the circle
sporadlcally appears as the philosO-
piers stone of mathematics. But it
is not by the antiquity of the problem
that circle squares are enticed. 1t is
ambition that spurred men on in an
Gent Greece and still spurs them on in
modern times to crack this primeval
mathematical nut.
Although the opinion is widely cur
rent in the professional world that
mathematicians are still busy with the
solution of the problem, the fact is that
for the past two hundred years the en-
deavOrs of many considerable mathe-
maticians have been solely directed to-
wards demonstrating with ~exactness
that the problem is insolvable; but it
was not until 1882 that Prof. Linde-
mann, of Konigsberg, succeeded in
furnishing a demOntration-and the
first demOnstration-that it is impOs-
sible by the exclusive use of ruler and
cOmpasses to construct a square that is
automatically exactly equal in area
to a given circle. The demonstration
naturally was not effected with the
help Of the old elementary methods;
for if it were, it would surely have
been accomplished centuries ago; but
methods were requisite that were first
furnished by the theory of definite in
tegrals and departments of higher
algebra developed in the last decades;
in other words, it required the direct
and indirect preparatory labor of many
centuries to make finally possible a
demonstration of the insOlvability of
this historic problem. 1t is, however,
not to be expected that this demOnstra-
ton will put an end to the race of circle
squarers.
Since the time of Archimedes, all
civilized nations that have cultivated
mathematics, have called the number
denotes how many times larger than
the diameter the circumference of a
circle is, pi-the Greek initial letter of
the word periphery. To compute pi,
therefore, means to calculate how many
times larger the circumference of a
circle is than it's diameter. This calcu-
lation is called 1.-The numerical rectifi-
cation of the circle"
The numerical rec~tification and
numerical quadrature of the circle
based upon the computation of the
number pi, are to be clearly distin-
guished from problems that require a
straight line equal in length to the cir
conference of a circle, or a square
equal in area to a circle, to be con-
structively produced out of its radius
or its diameter; problems which might
properly be called constructive recti-
fication" or constructive quadra-
ture." Approximately, of course, by
employing an approximate value for
pi these problems are easily solvable.
But to solve a problem of construction
in geometry, means to solve it with
mathematical eXactitude. If the value
pi were exactly equal to the ratio of
two whole numbers to another, the
constructive rectification would pre-
sent no difficulties. Yet as a matter of
fact, and as has been actually demon
strated, there do not exist two whole
numbers, be they ever so great, that
exactly represent by their proportion
to one another the number pi. Conse-
quentlyn a rectification of the kind just
| described does not attain the object
desired.
In the oldest mathematical work
that we possess we find a rule that tells
us how to make a square which is
equal in area to a given circle. This
celebrated work, the Papyrus Rhind of
the British Museum, was written, as is
stated in the work, in the thirty-third
year of King Ra-a-us, by the Scribe
Ahmes, accordingly in the period of
the two Hyskos d~ynasties, that is in
the period between 1000 and 100 be-
fore Christ; and Ahmes states that he
composed his work after the model of
| old treatises written half a thousand
years before. We find the ancient
Babylonians concerned with the prob
lem, and also the early Greeks, the
Romans, the Chinese and the Hindus.
The Hindus surpassed even the Greeks
in the exactitude of heirmathematical
labors, their result for pi being 3.1416,
while pi, really lies between 3.141592
and 3.141593.