The Newberry herald and news. (Newberry, S.C.) 1884-1903, 1886-03-03
(2Ne=576, add 100,=676=(26)s
(2?(s=529. add 200,=729=(2O2
(22)2=4S4, add 300,=7S4=(2S)2

N2=1. add 24O0,=240l-49)2
70 determine the square OF any
number between 25 and so find the
corresponding number below 23, and
augment.its square Ey the number of
hundreds indicated by its remotcness
from %. Or. more conveniently.
take the excess above % as hundreds,
and augment by the. square OF what
the number lacks Of 80.
Thus:
N3)s=((43-25} 100 plus (5o-t3)s
1900 plus 49=lS49.
Conversely: TO obtain Tie square
root of 1864. The root is plainly
between rs ard sc. The tens and
units figures indicate s. Therefore
the square root IF l7G4 is 5O-s=42.
1t is further observable that the
tens and units figures of perfect
squares Of numbers from 81 to SO
inclusive are the same us the tens
and units figures Of the squares OF
numbers from 10 to l inclusive.
Since multiplied by any number of
hundreds plus -, 80 or - gives an
exact number Of hundreds. it follows
that tile tens and units figures Of the
squares of numbers less than % rep
resent all the possible combinations
of figures in those orders Of units for
aN square numbers. The termina
tions of all perfect square numbers
are % in all viz, 90, OIL, Oh, %. l6,
p, ~, 25, 20. sc. Al. II, 40. Sc. 6i,
! CT, 60, 76, 81, S4, 50, 9G.
The following 'ulels then deduced:
TO square any number from 30 to
100. take twice the excess above 30
as hundreds, and augment by the
square Of what the number lacks of
100.
Thus:
(sys=2oo 5000 plus QoO-s9)a
= 1870 plus 121 1921
Conversely, the square root Of
3249: The rootis plainly between 80
and GO; the tens and units figures
indicate I; therefore the square root
Of 3249=50 plus 7=57.
For greater convenience it is noted
that in such q case as the square root
Of 1921 the root is so plus 80 or 100
Ill, and it is easier to use the latter
form. That is, if the root is in the
fourth quarter Of the hundred, SUI
tract the number indicated by the
tens and units from 100, and the diff
ference is the root. Thus the square
root of S2Sl=100-9=91.
To square any number from 100 to
200, take four times the excess above
100 as hundreds and augment by the
square Of what the number lacks Of
200.
TO square any number from l25 to
$50, take one-half the excess above
l25 as thousands and augment by
what the number lacks Of 250.
BY series Of steps OF this charac-
ter the author gives methods for
squaring higher numbers and con-
versely for obtaining their square
roots. A choice Of methods is alsn
indicated. The facility which was
obtained by such means was d. fly
illustrated on the blackboard by the
author. who in a few seconds per-
formed such exploits as raising 5 10
the 18th power, and then showed in
detail theprocesses which he had men
tally executed. The paper sets forth
the reason for each rule, deducing it
from the usual binomial theorem,
with almost obvious simplicity.
The demonstrations were received
by the section with hearty applause.
In response to an inquiry, Mr.. West
cott stated that he had been very
successful in teaching this method in
classes, about q tenth Of lis pupils
becoming rapid experts in the meth
ods of solution, which were especially
usefat in handling quadratc equa-
tions, and determinining at a glance
whether a given number is or is not
perfect square.-Scientifc american.
*Read at the recent meeting of the Amcri-
can Society, 1888.