Note: Descriptions are shown in the official language in which they were submitted.
CA 02217809 1997-12-08
Description
First draw a circle, angle, and bisect
arcs. Place disk with holes over circle. The
needles find the points; the buttons thread
the needles positioning disk; gets clamped;
unthreaded; insert pegs; attach central and
chordal arms. The nipple finds the groove;
rotate central, slide nipple, rotate, slide,
mark the spot. Do other side. Accessories aid
operation. Special parts for trisecting very
small angles are to 1/6 scale; rest to 1/3;
except details to full.
March 17th 1997
* special thanks to business analyst
Mary ~llen Heidt of S.I.D.C.O.
(Southern lnterior Development Cor-
poration of the Okanagan-~imilkameen)
in helpir-g to reduce description of
the ~anipulation of the device to less
43
CA 02217809 1997-12-08
than 75 words.
~pecifications
radius = 15 centimeters
material - solid wood
model ~ T-1000 series or T-1001
Thooem Models
T-1001 solid material (as sho~m)
T-2001 clear see-through (not shown)
T-3001 clear see-through (not shown)
~-1001 prototype (as shown)
T-1002 working model (not shown)
M.W.M. November 26th 1997
44
CA 02217809 1997-12-08
Geometry
The tools of fundamental geometry or
basic construction are well known to be a
simple compass and an unmarked straight-edge.
With the simple tools it is possible to draw
a random angle, and then divide it into two
equal angles. ~he method of construction goes
as follows:
1) - compass a circle
2) - strai~ht-edge a radius
~) - str3ight-edge another radius to form
a random angle
4) - compass arcs from those two points on the
circumference to find a point of inter-
section which is the bisect point
5) - connect the bisect point to the center
point of the circle or vertex of the
angle with the bisect line
~ CA 02217809 1997-12-08
iv
ii O ,~ iii
~ ,
The double arc bisect method is
absolutely precise. The proof begins by
connecting the bisect point to the two
points on the circumference. This forms
an equilateral quadrilateral because all
four sides are radii.
~ CA 02217809 1997-12-08
iv
4 ~iii
At the same time two triangle are formed
which are exactly the same via the S.S.S.
triangle congruency principle. Therefor angle
ii-i-iv is the same as angle iv-i-iii.
The double arc bisect mehhod can be proven
in another way also, because an equilateral
quadrilateral is a parallelogram which ha~,the
property of ppposite angles being equal. Therefor
the two triangles are the same via the S.A.S
triangle congruency principle.
It is also possible to prove something else!
CA 02217809 1997-12-08
V i ii
~.~ ~
"_
We have already proven that angle "y"
equals angle r'y'l. Now when we connect points
ii-iii with a chord, the bisect line intersects
the chord at point v. The triangles i-ii-v and
i-iii-v are exactly the same via the S.A.S.
triangle congruency principle. ~herefor line
segments ii-v and v-iii are equaI to each other!
This fact can be used if the construction
were to continue.
CA 02217809 1997-12-08
This construction is long and complicated
but all points are clearly defined. ~he method:
1) compass a circle
2) str~ight-edge a radius
3) another radius forms a random angle
4) connect those two points on the circumference
with the chord of an isosceles triangle
5) put the double arc bisect method into play
and the bisect line intersects the chord at
a c~rtain point, the initial adjustment point
6) compass an arc from it finds two points on the
bisect arcs
7) connecting those two points to the vertex of
the angle finds two new points on the chord,
the second adjustment points
8) compassi~g arcs from each of them finds two
new points on the bisect arcs further out
9) connecting them to the vertex of the angle
finds two new points on the chord, the third
set of adjustment points
10) compassing arcs from each of them finds two
new points on the bisect arcs a wee bit
further out
11~ connecting them to the vertex of the angle
apparently trisects the angle according to
a measurement with a protractor
12) a geometrical proof follows
/
Most of this has already been proven.
The bisector of the chord can also be
proven to be perpindi cular to it. But
this is not the feautre we build upon.
'I'hose two isosceles triangles can be shown
to be exactly the sa~e thanks to S.S.~.
/2
k- ~'- z\
~x X~
\/
.,
.. /
~.
.,...~
~o then angle z e~uals angle z
And T.~e already have angle x equal to x
And there is the axiom 'l~he sums of equals
added to equals are equal. !l
tS"
So those two angles are equal
And those two triargles congruent via S.A.S.
/i7L
, /
i ,
r
~r
S~ ff'
~, .
~ , ~
So then the two outside angles have been
shown to be the same, and this carries on
~C
I ~ A. ~ . A. a t~ri a-~- gle c ongruency principle ?
, l;
Vd ~ 7 ~
'l /'
'~ ''.
i
,.
,
..
If so.
y '~
~ s
And w~ got to go through that all again.
/~
1~ 1
ll
And through it again.
17
!
'i ,
'i f
;s~ ".~
,~ ,.
~o get to here, where those two angles have
been conclusively show~ to be e~ual for sure.
7 ~
If we could prove that the center angle
is e,ual to one of the outside angles
that would prove that all three are equal.
~o just consider, if you will, the distance
from point xiv to point xviii, or the following
dotted line
21
~' f/
t \ ~
~.
e
_, ,Ji'
.. ........
If that dotted line is a radius, then we
would have an equilateral ~uadrilateral,
~hich has a property of a diagonal seperating
it into two congruent triangles, which would
prove that the ce:l~ter angle is the same as
the one on the outside.
CA 02217809 1997-12-08
This is ;ot a blank page. Above
is a drawing of a circle, angle,
iso cles triangle, double arc
bisect & triple adjustment with
a double pointed compass. The
sa.rne compass mea~ures a certain
two points to be a radius apart.
CA 02217809 1997-12-08
Analt~sis
A compass and unmarked str-,ight-edge
construction can be plotted on the Gartesian
coordiaate system with (x,~) points and lines
in the y=mx + b format and circles in the
x2 + y2 = 1 format. ~ince there is an old
and well known meth~d to trisect a right
angle of 90~ we can figure out the slope of
a 30~ angle.
m = .57735~
Now we can run the new method through
the equations to see if the third adjustment
fou~d the slope of a ~0 angle.
m = ,579876
It came up a few thousandths of a slope
in the ~0~ range short. Therefor the triple
adjustment trisect method is not absolutely
precise. Then one wonders if another adjustment
would find it, but after the fourth we get
m=.577844 which is still a lit.le bit short.
24
After the fifth adaustment the slope
m=,577447 which is still short. After the
sixth adjustment the slope m=.577369 which
is still short. After the seventh adjustment
the slope m=.577354 which is still short but
shows that the rnethod is bee bopping towards
the ~0~ line, or rather the method is honing
on in on it.
It takes an infi~ite number of adjustments
to trisect an angle precisely. ~owever a grid
can only be made so small, only so man~ dots
per inch. Meanwhile the rlew method of const-
ruction can be programmed to infinity with
simple algabraic instructions.
2~
CA 02217809 1997-12-08
Perhaps the new method of con~truct-
io~ could be put into a computer program
or graphic displav device or plotter, but
that is beside the point of this letter.
T~!iih just a simple compass and unmarked
straight-edge an~ angle can be trisected
according to a naked eye observation, and
the thooem could never hope to accomplish
more than that.
3o
CA 02217809 1997-12-08
~ ~ top view
j~ ,
.~ .
, ~ . . ,
r -~ 5 ide view
_ _ L I _
The double arc bisect mehhod got put into a tangible form
in the angle bisecting tool. Basically it is composed of
two disks hinged together at their circumferences.
. .
~imilarly the triple adjustment trisect method can be put
into a tangible form in the thooem. Basically it is com-
posed of two disks. The one has a nipple at its center.
The other is hinged to a central arm, and hinged to a
chordal arm which has a slot for the nipple a~d a slot
for alignment purposes.
