Improvements in the Method of and Instruments for the Drawing of Equilateral Triangles, Squares, Pentagons, Hexagons, Heptagons, and Octagons.
Year: 1892

page_text: Date of Application, 14th May, 1892
Complete Specification Left, 22nd Nov., 1892-Accepted, 31st Dec., 1892
PROVISIONAL SPECIFICATION.
Improvements in the Method of and Instruments for the Drawing of
Equilateral Triangles, Squares, Pentagons, Hexagons, Heptagons,
and Octagons.
I, WILLIAM RADCLIFFE, of the Schoolhouse, Andreas, Isle of Man, School-
master, do hereby declare the nature of this invention to be as follows:-
The said invention consists in improvements in the method of and instruments
for the drawing of equilateral triangles, squares, pentagons, hexagons, heptagons
and octagons.
In drawing equilateral triangles, squares, pentagons, hexagons, heptagons, and
octagons without the aid of mathematical compasses or protractors I use an
instrument which may be conveniently described as a flat rule having 8 consecutive
ruling straight edges in the same plane, such rule being capable of being used upon
either flat surface. The relation of the said edges to each other and the manner
of obtaining them I will now describe by reference to the accompanying lettered
plan of this invention.
The
The edge A H forms with the edge A B an angle of 90 degrees, and with the
edge H G an angle of 108 degrees. The edge GF forms with the edge H G an
angle of 135 degrees. The edge BC makes with the edge A B an angle of
120 degrees, and with the edge DC an angle of 60 degrees. It follows from a well
known geometrical theorem that the edge D C is parallel to the edge A B.
edge FE forms with the edge ED an angle of 128 degrees. The said edges
may be varied indefinitely in their respective & relative lengths whenever such
can be done without altering the sizes of the angles respectively formed by any
two of the said edges which sizes I have already named. By preference, however,
I make the length of the edge A B four times the length of the edge A H and
three times the length of a perpendicular G I drawn from a point G to the
edge A B and thus I fix the length of edge H G. By preference, also, I make the
parallel edge D C the same distance from edge A B (as measured by a perpen-
dicular DL) as is point H in edge A H from point A in edge A B. It follows
from this that a line joining point H with any point in edge DC will form one and
the same straight line with edge DC. By preference I make the termination of
the edge G Fat point F in the line H, and point G having been already fixed,
I thus fix the length of edge G F. By preference I also make point E the same
distance from edge A B (as measured by a perpendicular E K) as is point G from
edge A. B, and thus 1 fix the length of edge F E. Again, by preference, fixing
the extremity of edge ED at point D in the line HCI thus fix the length of
edge E D. And point D having been fixed I thus fix the length of edge DC.
This instrument may be made in any convenient size of wood, metal, or other
material of suitable thickness and may be formed by any of the well known
methods (suitable to the material used) for giving a particular outline to a flat
surface.
For the sake of convenience 1 inscribe the numerical values in degrees of the
respective angles in, or near, the angles to which they refer; and for the same
reason I also inscribe some suitable scale or scales of inches along the edge A B
of the instrument.
Now, the way in which I use this instrument for the purpose of describing the
aforesaid geometrical figures is as follows:
Each angle formed by any two consecutive sides of an equilateral triangle is
60 degrees; of a square 90 degrees; of a pentagon 108 degrees; of a hexagon
120 degrees; of a heptagon 128 degrees; of an octagon 135 degrees.

page_text: Now suppose, for illustration, I wish to make a pentagon. Using the long edge
of this instrument as a rule, I draw a base line of any desired length. Each
interior angle of a pentagon being 108 degrees I proceed to make an angle of
108 degrees at either extremity of the said base and on either side as may be most
convenient. I obtain the angle of 108 degrees in one of the four positions aforesaid
by making either the edge A H or the edge H G coincide with the said base so
that the point H will coincide with the point forming the extremity of the base at
which it is desired to form the said angle, and by using edge H G or edge A H, as
the case may be, as a rule to draw the line which forms with the base the said
angle. I now by means of the inscribed inch scale, or scales, measure the base
length and make the newly formed line, just mentioned, of equal length with it.
I have thus obtained a side of the pentagon. Taking this side as a base I repeat
the process, and thus form another side, and so go on until the pentagon is
complete. The other flat surface may be used for the same purpose. And it is
evident that any of the geometrical figures aforesaid may be formed in a similar
manner, using the angle proper to the particular figure to be made.
Dated this 10th day of May 1892.
WILLIAM RADCLIFFE.
COMPLETE SPECIFICATION.
Improvements in the Method of and Instruments for the Drawing of
Equilateral Triangles, Squares, Pentagons, Hexagons, Heptagons,
and Octagons.
I, WILLIAM RADCLIFFE, Schoolhouse, Andreas, Isle of Man, Schoolmaster,
do hereby declare the nature of this invention and in what manner the same
is to be performed, to be particularly described and ascertained in and by the
following statement :-
In drawing equilateral triangles, squares, pentagons, hexagons, heptagons and
octagons without the aid of mathematical compasses, or protractors, or "protractor-
rules," I use an instrument the manner of constructing and of using which, I will now
describe with reference to the annexed drawings.
Figure 1 is a plan of the instrument showing its essential component parts, and
Figures 2, 3, and 4 are plans of modifications of the same instrument.
The instrument of which Figure 1 is a plan may be conveniently described as a
rule, flat on both sides, having 8 consecutive, ruling straight.edges in the same
plane.
In Figures 1, 2, 3, & 4 the line A. H forms with line A B an angle of 90 degrees,
and with line H G an angle of 108 degrees.
In these 4 figures also, the line BC makes with the line A B an angle of
120 degrees and with the line D C an angle of 60 degrees.
It follows, from a well known geometrical theorem that the line D C 18
parallel to the line A B. In Figure 1 the line H G makes with the line G F
an angle of 135 degrees, and the line F E makes with the line E D an angle of
128 degrees.
In Figure 2 the line H G makes with the line G F an angle of 135 degrees,
and the line M E makes with the line E D an angle of 128 degrees. In
Figure 3 the line H G makes with the line G F an angle of 1284 degrees, and
the line F E makes with the line E D an angle of 135 degrees.
In Figure 4 the line H G makes with the line G F an angle of 128+ degrees,
and the line M E makes with the line E D an angle of 135 degrees.
The edges represented by the said lines may be varied indefinitely in their
respective lengths, in either Figure 1, 2, 3, or 4, whenever such can be done
without altering the sizes of the angles respectively, formed by the above specified
pairs of lines, the sizes of such angles having also been specified above.
By preference, however, I make the length of the edge A B four times the length

page_text: of edge A. H and about three times the length of a perpendicular G J drawn from
point G to A B. Thus, also, I fix the length of HG in Figures 1, 2, 3 & 4.
Again, by preference I cause the parallel D C to be the same distance from A. B
(as measured by a perpendicular D L upon A B) as is point H in A H from
point A in A B. It follows from this that a line joining point H with any point
in D C will form one and the same straight line with D C. By preference, also,
I fix the termination of line G F in Figures 1, 2, 3 & 4 at point F in line H C and
thus I fix the length of G F in each of the four cases.
In Figures 1 and 3 the line F E is drawn from point F in H C (the extremity
of G F to a point E the same distance from A B (as measured by a
perpendicular E K) as is point G from A B.
In Figures 2 and 4 the line M E is drawn.from a point M (situated in line H C
in such position that the angle G F M is exterior to the triangle H G F) to the
point E, the position of which is fixed in the same way as it has been fixed in
Figures & 3, and mentioned above.
In Figures 1, 2, 3 & 4 I fix the termination of line E D at point D in H C and
thus I fix the length of E D in each of the four cases.
Thus also I fix the length of DC in each of the four cases.
This instrument may be made in any convenient size of wood, metal, or other
material of suitable thickness and may be formed by any of the well known
methods (adapted to the material used) for giving a particular outline to a flat
surface.
For the sake of convenience in using this instrument I inscribe the numerical
value in degrees of the respective angles in close proximity to the angle to which
the said numerical value refers. I likewise, for the sake of convenience, inscribe
some suitable scale or scales along one or more of the edges of this instrument, the
edge to which I give preference for this purpose being the long edge A B.
Now it is a mathematical fact that the interior angle formed by any 2 consecutive
sides of an equilateral triangle is 60 degrees; of a square 90 degrees; of a
peutagon 108 degrees; of a hexagon 120 degrees; of a heptagon 128 degrees;
of an octagon 135 degrees. These are the angles (whose numerical values are
inscribed in proximity to them) which form the interior angles of this instrument
as shown in Figures 1, 2, 3 and 4.
The method of using the instrument is therefore as follows:-
Make a base line of any desired length, then form at either extremity of the
base, on either side of it, the angle proper to the geometrical figure intended to be
made that is to say, for an equilateral triangle an angle of 60 degrees; for a
square an angle of 90 degrees; for a pentagon an angle of 108 degrees; for a
hexagon an angle of 120 degrees; for a heptagon an angle of 128 degrees; for
an octagon an angle of 135 degrees.
Having made the requisite angle the next thing is to make the newly formed
angle-arm of the same length as the aforesaid base line. Two sides of the
geometrical figure to be made will now have been formed, and the remaining sides
can be formed by treating the second side formed as a new base, and repeating the
above process, taking care that the requisite angle is formed at the proper extremity
and on the proper side of the new base.
The angles are formed by placing the instrument in such a position that one of
the two edges containing the angle coincides with the base line while the other edge
runs away from the chosen extremity of the base and on the chosen side of it.
Draw a line from the extremity of the base along this edge and the angle is
formed.
It is manifest that the instrument may be used, for the purposes above described,
with either surface uppermost.
Having now particularly described and ascertained the nature of my said invention,
and in what manner the same is to be performed, I declare that what I claim is:
1. A rule flat on both sides, having 8 consecutive, ruling straight edges in the

page_text: same plane, and containing 6 interior angles of the following magnitudes in degrees,
viz. 90, 108, 135, 128, 60, and 120, for the purpose of describing the geometrical
figures above set forth."
2. The combination and arrangement of the sides and the arrangement of the
angles illustrated and described in reference to Figures 1, 2, 3, and 4.
3. The combination and arrangement of the sides and the arrangement of the
angles obtained by placing the instrument in such a manner that the reverse side
or surface to that indicated in Figures 1, 2, 3 & 4 is uppermost, whereby the angle
at the left extremity of the long edge would be changed from 90 degrees to
120 degrees and the other angles correspondingly changed.
Dated this 19th day of November 1892.
WILLIAM RADCLIFFE,
Applicant.