The Manitowoc pilot. [volume] (Manitowoc, Wis.) 1859-1932, 1881-02-03
:Gouilowoc pilot.
THURSDAY, FEBRUARY 8, 1881.
4 MATHEMATICIAN PARADOX.

Mathematics is usually regarded as
sid science With principles fixed and
innutable, and conclusions definite aud
rnauosbiouabIc. The fantastic capers
Which It cuts up with the representation
q Ala6ama according TO tables prepared
l. Tie superintendent Of the census, in
r. alton to apportionment Of representa-
tt5s viii make many believe that math-
enatics IS allied with necromancy and
tat It has enjoyed a reputation for fzity
aad certainty which It has 10 no way de-
served.
The superintendent Of the census pre-,
pared d table showing the number Ct
representatives to which each state vas
entitled on an aggregate membership Of
ir5n 293 to 397 both inclusive. The
rlognod employed ior ascertainiug these
lumbors Is to find the basis Of represen-
iasion by dividing the representative
population by the final number Of mem
lers. This basis applied 70 the popula-
iion Of each state gives the number Of
representatives to which It IS entitled,
I,27iug a fraction over. Tae sum Of The
representatives accorded to each state by
flis even division Is less than the whole
number Of representatives on which the
calculation is made The states having
ihe largest fractions are given an addi-
4ioual representative until tie whole
number Of representatives, of which the
house il "a lie composed Is assigned
According to ihi5 plan The state Of Ala
1ama, wien the house numbers 293 4 >
CTS, hes Z representation Of 7. When
Inc house numbers 239 it gains one, and
j. thus entitled to s. When the number
III representatives, is 800, Alabama loses
the representative gained On basis Of
90 members. To fully appreciate the
anomaly we must consider that when the
whole number Of representatives IS in
creased from 800 to 500, the number to
which Alabama is entitled is reduced.
When the representative population re-
oaired for one member IS reduced the
representation Of the state is reduced, a
jet opposed to all principles of proper
ton. In other words, the smaller the
aiyi4or the less the quotient. It is quite
apparent that this anomalous condition
results from the fraction of the represen-
fa5iYe population left unf6presented by
a. even division. On basis Of 800, on
n even division, Alabama is entitled to
z representatives, and the whole number
of representatives apportioned on such
division is 277, leaving y TO be assigned
T. the larger fractions. OUnitting Neva-
q. in the calculation, which is entitled to
q representative, not on its population,
Iut on q constitutional provision, the
smallest fraction entitled to an additional
representative is f0Und In Alabama; that
j. when the 299 members are gned by
fho addition Of y members to the 277
apport.cned by even division, Alabama
receives the 22nd, and no state with q
] IS fraction is entitled 60 an additional
7lam5cr because the number 299 is com
Tlala. The Iowcstre remainder, Alabama's,
j. l0G. $50. Texas and Illinois stand
next, having respectively LOG, $50, and
115.G3I. On basis of 800, Alabama on
an even division Is still entitled to rep
T1seutgtives, but now the resulting frac-
zion or remainder is relatively different.
Alabama has 1I0,839 unreprcsented;
'Tczas, III, $50; and Illinois, 1859.
n assigning the 20 additional members,
Tie number of representative on even
division being now 80. Illinois and
Texas take precedence and the 20 are
exhausted before Alabama is reached. so
that she has only the members to which
sae is entitled by even division.
50 peculiar q result called forth an ez-
planation from General Walker. His
demonstration Of the paradox IS easily
understood, and is ia substance as fol-
lows: TIE number Of Inhabitants re
CURED for q representative. on q basis Of
so members IS SOL less then on q basis
rf2G9. On m even division the repre-
santativas Ct Illinois. Texas and Alabama
oro tbe same respectively as on even di-
vision on q basis Of 231, VIZ: 7, 8, IS; as
the basis Of representation has decreased
55I for each memhsr, the unraPressnted
faction increases in each stata different
l., in Illinois 55lzl8; in Texas 551x9; in
Alabama 551z7; adding each Of these to
tie unrepresansed fraction on a basis Of
90 members, and We have for Illinois
JIG, 500; for Texas al $50; for Alabama
ll0S39. Thus as Alabama stands be
low 20tb, the ordinaI corresponding with
tlo number Of additional representatives,
assigned, she fails to get an additional
Representatives on her fraction.
-.sE--.---z-zs----.-s..----
THE efforts Of the English government