The Hocking sentinel. [volume] (Logan, Ohio) 1871-1906, 1885-03-12
'sTHAT LADDER. %
Climbing it by TrfgOncmetry.


10 THE HOOKING SENTINEL
DEAR EDITOR II have read. il
your valuable paper on severa
occasions, of peculiar Quirk il
I, the minds cf the County Boart
Of School Examiners, with refer
ence TO a certain class of mat ha
matical problems, popularly
called Ladder Questions"
Permit me 10 call The attention
Of our worthy Board of Exami
ners to the we'll settled fact, that
The human mind, dwelling Ions
upon any single subject, becomes
biased, upon the same and inca
pable of proper di scrimination
The class Of questions referred
to as Ladder questions" are al.
most as numerous as the sands
on the sea shore, or the stars in
the heavens, and all to a greater
or less extent involve the multi
tudinous variations of the reia-
tion and proportion of lines and
angles.
TO our mind the whole catego-
ry of problems belonging to the
class named, are clearly outside
the pale of true arithmetic, al
though some Of them are s"lva-
BILE by certain set formula de
drced from the higher mathe-
matics, but when they are SO SOL
vable the formula must be strict
ly applied.
The few questions of this class
susceptible of arithmetical solu-
tion as above indicated, are con
fined to such as contain one right
angle, two sides being given, or
in other figures susceptible of
division in two or more tri-
aHgles,each of which contain one
right angle and two known sides.
So long as the Examiners see
proper 10 coniine themselves to
the last named division, the
problem may be admitted. But
at The last examination held by
the Board The following prob
lem was given the class for solu-
ton:
Two ladders are standing in
the street with their bases 90 feet
apart inclined toward each 9TH
er forming an angle of As. de-
grees where they meet at the
top. Find the length of each
ladder,"
This problem does not CONTAIN
sufficient data for an arithmetic
al solution, nor can it be arrived
at by any combination Of the sci
ence of arithmetic.
Mechanically speaking, the
gwen data is the length of the
chord. 20 feet, and the included
angle 45 degrees.
Without pauslng to give def
milton we will state that unless
some further data is given, as for
inst ance, the length Of the cosine.
il is impossible 10 solve the prob
|em without the use of Trigo-
Arithmetic cannot deal with
angles sines, cosines, CO
versed sines and tangents, nor
can it discover The relation they
bear 10 each other.
To make clear a complete So
luton of the problem, it would
be necessary TO give a diagram
which l presume you can not
10, in the paper, but l will
Try to do SO without, but | have
some doubts as to my ability to
make it clear.
The chord is straight line
connecting the extremeties of an
arc. and is consequently twice
he length of the sine of one half
he angle.
Reasoning from this we have
he sine equal to one half the
.hord, or 10 feet. and the angle
opposite the given side or sine,
equal TO one half of 45 degrees.
fis: 90 so and the angle at the
of the sine with the
:osine equal to 800. Omitting
figures, by means of natural
'ines, and the following formula,
viz:
"As the sine of the angle
opposite the given side is to the
line of the angle opposite
he required side, so is the given
'ide 10 the required side,'' we
ind the length of the ladder to
,e 26.13l feet.
But the same result may he
lrrived at by using The chord as
.ivcn by the Examiners and the
whole given angle, viz: 40. The
adders being the same length il
5 evident that the angle formed
~ the given chord and the lad i
lers must he the same. then 1902
minus 450 equals 1859 divided by I
equal SO so. Then as the sine /
If 450 equal 10 .707lO6S is to sine
# SO SO equal TO .923S795,
o is the chord 20 feet. 10 the i
ength of the ladder, 26.13l feet.
There can be no reason of
COMPLAINING ofthe mania of the
for ascertaining the
en.th of every ladder that they
an hear of or cO..ceive of. but