Improvements in the Method of Dividing Angles into any Number of Equal Parts, and in Appliances for Effecting the same.
Year: 1894

page_text: Date of Application, 5th Dec., 1894
Complete Specification Left, 4th Sept., 1895-Accepted, 19th Oct., 1895
PROVISIONAL SPECIFICATION.
Improvements in the Method of Dividing Angles into any Number
of Equal Parts, and in Appliances for Effecting the same.
I, LOTHAR VON KÖPPEN, of Kraieling 61 Munich, in the Kingdom of Bavaria
and Empire of Germany do hereby declare the nature of my said invention to be
as follows:-
My invention relates to an improved method of dividing angles into any
required number of equal parts, by means of an improved mathematical
instrument for effecting the same.
And in order that my said invention may be particularly described and
ascertained, reference is hereby made to the accompanying drawings, in which
similar letters and figures of reference indicate corresponding parts.
Fig. 1 illustrates the principle upon which my invention is founded, and
Fig. 2 illustrates the form of instrument used in applying my invention to the
division of angles.
Referring to Fig. 1: Let B A C be the given angle to be divided. Taking A
as a centre, and any radius, as A a; cut off equal parts A a and A b on the
lines A B and A C. Draw through the points a and b the lines a y
parallel to AC and A B.
and b
Next, beginning at a b, describe arcs, as shown, at equal distances apart, radially.
Then transfer the lengths of the arcs cut off by the lines a y aud b to the
respective arcs; join these points, forming thereby the curved lines a¹ c¹, a³ c³,
al clo, a cl, a¹³ clo, a c; also, in like manner, the lines b d¹ to 24 d. The
points represent the division points for trisection; m m² m² for four part
sections ;nn' n² n³ for five part sections; p p¹ p² p³ ³ for six part sections; and
ro¹ po² på pź po for seven part sections.
Referring to Fig. 2: The two plates ss2, in the form of sectors, and of equal
dimensions are pivoted together at t, the plate is made of transparent material,
and capable of sliding upon the plate s¹. The rule u is also pivoted at t, one edge
being radial to the centre t.
The geometrical diagram shown in Fig. 1 is drawn upon the two plates s¹ s³, in
such a manner that the plate s¹ is provided with one of the two systems of curved
lines, and the plate with the other system, as shown in Fig. 2.
The upper plate is moved by means of the handle r, upon the lower plate, until
the required angle is obtained between radial lines B and C. The movements of
the upper plates do not alter the designs, but intersection points, where the
corresponding curves cut one another, change their position according to the given
angles to which the instrument is adjusted; these intersection points furnishing the
required points of division.
The angle to be divided may be of any size whatever, and the given angle may
be divided into any number of equal parts.
In using the instrument shown in Fig. 2, let A B C be the given angle to be
divided into any number of equal parts. Set the instrument so that one edge of
the under plate s coincides with one sile, A, B, of the angle; the apex A
coinciding with the pivot t. The upper plate is similarly placed with respect to
the other side A, C, of the angle. In this position of the instrument the inter-
section points where the curves of both systems cut each other, furnish the division
points of the angle, as required.

page_text: By means of the rule u, applied to the corresponding points of intersection, lines
may be readily drawn corresponding with the points of intersection.
In Fig. 2 the angle A B C is shown divided into three equal parts, the rule »
being set to the point for trisection, that is to say, B A D is equal to one third
of BA C.
It is to be regarded as a geometrical deduction that the edge D of the rule u
must pass through the points of the intersections forming multiples of three; as,
for instance, through the points 9 and h, the first being the intersection point for
the division of the given angle into 6 equal parts, and the second being that for the
division of the angle into 9 equal parts.
Dated 5th December 1894.
Improvements in the Method of Dividing Angles into any Number
of Equal Parts, and in Appliances for Effecting the same.
I, LOTHAR VON KÖPPEN, of Kraieling, 61, Munich, in the Kingdom of Bavaria
and Empire of Germany, do hereby declare the nature of my said invention and
in what manner the same is to be performed, to be particularly described and
ascertained in and by the following statement :--
My invention relates to an improved method of dividing angles into any required
number of equal parts, by means of an improved mathematical instrument for
effecting the same.
And in order that my said invention may be particularly described and
ascertained, reference is hereby made to the drawings filed with my Provisional
Specification, in which similar letters and figures of reference indicate
sponding parts.
Fig. 1 illustrates the principle upon which my invention is founded, and
corre-
Fig. 2 illustrates the form of instrument used in applying my invention to the
division of angles.
Referring to Fig. 1: Let B A C be the given angle to be divided. Taking A
as a centre, and any radius, as A a; cut off equal parts A a and Ab on the
lincs A B and A C. Draw through the points a and b the lines a y and b =
parallel to AC and A B.
Next, beginning ut ab describe ares, es shown, at equal diterpart, elially.
Then run the length of the ares cut off by the line and to the
respective area; join these points, forming thereby the curvel lines
ap; also, in like manner, the line to
points represent the division points for triscation;
Sections;
and
The
for four part
for five part sections; for six part sections;
for seven part sections.
Referring to Fig. 2: The two plates as, in the form of acctors and of equal
dimensions are pivoted together at t, the plate is made of transparent material,
and capable of sliding upon the plate s'. The rule u is also pivoted at t, one edge
being radial to the centre t.
The geometrical diagram shown in Fig. 1 is drawn upon the two plates s' s², in
such a manner that the plate s' is provided with one of the two systems of curved
lines, and the plate 2 with the other system, as shown in Fig. 2.
The upper plate is moved by means of the handle v, upon the lower plate, until
the required angle is obtained between the radial lines B and C. The movements
of the upper plate of do not alter the designs, but intersection-points, where the
corresponding curves cut one another, change their position according to the given

page_text: angles to which the instrument is adjusted; these intersection points furnishing the
required points of division.
The angle to be divided may be of any size whatever, and the given angle may
be divided into any number of equal parts.
In using the instrument shown in Fig. 2, let A B C be the given angle to be
divided into any number of equal parts. Set the instrument so that one edge of
the under plate s¹ coincides with one side, A, B, of the angle; the apex A
coinciding with the pivot t. The upper plate s² is similarly placed with respect to
the other side A, C, of the angle. In this position of the instrument the inter-
section points where the curves of both systems cut each other, furnish the division
points of the angle, as required.
By means of the rule u, applied to the corresponding points of intersection, lines
may be readily drawn corresponding with the points of intersection.
In Fig. 2 the angle A B C is shown divided into three equal parts, the rule u
being set to the point i for trisection, that is to say BA D is equal to one third
of BA C.
It is to be regarded as a geometrical deduction that the edge D of the rule u
must pass through the points of the intersections forming multiples of three; as,
for instance, through the points 9 and h, the first being the intersection point for
the division of the given angle into 6 equal parts, and the second being that for the
division of the angle into 9 equal parts.
Having now particularly described and ascertained the nature of my said inven
tion, and in what manner the same is to be performed, I declare that what I
claim is :-
1. The improved method of dividing angles into any number of equal parts,
substantially as herein before described and illustrated by the drawings.
2. The application and use of the improved appliances for dividing angles into
any number of equal parts, substantially as herein before described and illustrated
by the drawings.
3. The improvements in the method of dividing angles into any number of equal
parts, and in appliances for effecting the same, substantially as herein before
described and illustrated by the drawings.
Dated 3rd September 1895.
GEO.
DOWNING,
Quality Court, London, W.C., Agent.