Instrument for Describing Conic Sections and for Trisecting Angles.
Year: 1899

page_text: Date of Application, 26th June, 1899-Accepted, 23rd Sept., 1899
COMPLETE SPECIFICATION.
Instrument for Describing Conic Sections and for Trisecting
Angles.
I, WILHELM ZIETHEN, of Schlachtensee, near Berlin, Germany, Engineer, do hereby
declare the nature of this invention and in what manner the same is to be per-
formed, to be particularly described and ascertained in and by the following
statement:-
The object of this invention is to provide an instrument by means of which
ellipses, hyperbolas, and parabolas can be described and also angles can be trisected.
The instrument comprises a rhombus shaped horizontal frame formed of four links
jointed together, said frame having connected with it a diagonal rod and a guide
rod, which rods carrying a marking style so as to be displaceable on them both.
In the accompanying drawings Fig. 1 is a diagrammatic view mathematically
shewing the description by the instrument of an ellipse. Figs. 2 and 3 the same for
the hyperbola and parabola, and Fig. 3ª the same for the trisection of angles. Fig. 4
is a plan view of the instrument when it is used for describing an ellipse and Fig. 5 a
side view of the same. Figs. 6 and 7 respectively a plan and side view when used for
describing an hyperbola, and Figs. 8 and 9 similar views when used for describing a
parabola.
"
The invention is based upon the following, geometrical theorems.
=
1. As regards the ellipse (Fig. 1):-Let FF be the foci of an ellipse; from Fi
describe a circle with the radius FD; let EDGF be a rhombus the points EG of
which lie upon the circle described from F..: The point D lies upon the circle
described from F1. The triangle DFP, is isosceles because the diagonals of a rhombus
come at right angles to one another halving themselves and the angles of the rhombus,
FIPPD
therefore DP PF. The line DF consists of FP and PD, therefore FID =
= PF. This demonstrates that point P is a
and F¹D PF PF because DP
point of the ellipse having the foci F,F¹, and the radius vectors PF and PF¹. The
diagonal EG of the rhombus EDGF is a tangent of the ellipse at point P. Supposing
the points F1 and F are fixed and the point D moves further on the circle of which it
is a point then the rhombus will be turned and its parts displaced on its joints. The
diagonal EG always contacts with the ellipse at one point, that is, encloses the
ellipse.
=
2. As regards the hyperbola (Fig. 2):-Let FF¹ be the foci of an hyperbola; from
F¹ describe a circle with the radius F¹D; from F, a circle with the radius FE or FG;
that
D of the jointed rhombus DEFG lies upon the circle described from F¹; E and G
upon that described from F. The triangle DFP is again isosceles so
5 DP PF. F'D=FP-DP=F¹P-FP. The difference of the radius vectors is con-
stant. The diagonal of the rhombus will therefore, analogous to the construction of
the ellipse, upon the further movement of the point D on the circle, enclose an
hyperbola with the radius F¹D.
3. As regards the parabola (Fig. 3):-If instead of the guide circle with the
0 radius DF of the ellipse and hyperbola, a circle is used of infinitely large radius,
that is, a straight line, then the line MN perpendicular to DP will be that which
must be put in the place of that guide circle. Then, upon the displacement on the

page_text: straight line MN of the point D of the jointed rhombus DEFG, the diagonal GE will
enclose a parabola with the focus F and the line MN as directrix. DPF is again an
isosceles triangle, PD - PF so that point P must always be at the same distance
from the fixed turning point F as from the line MN.
4. As regards the trisection of angles (Fig. 3ª):-Let MOX be the angle to be
divided and side MO be lengthened past O and at O a circle described of any con-
venient radius OP. This circle intersects MO or its extension in P and P¹; further
let an angle of 45° be drawn at OX and O inwards to cut the circle described from 0
in F and F¹. On P as a centre describe a circle with the radius PP and draw a
rectangular hyperbola having the foci FF and one point of which is P, finally, halve
at B the line RP the point R of which is the point of intersection with the circle
described from P, then BOX=½ MOX. Further to shew this let Fig. 3ª be ex-
tended by drawing PR until it intersects the lines OX and OY the latter being
perpendicular to OX. The points P and R are points of the same hyperbola, the foci
of which are FF1 and whose asymtotes are represented by the lines OX and OY. Now
the hyperbola has the property that the parts cut off, by the extension of a chord
through the asymptotes, are equal to one another. As in the construction shewn
PB BR and SP = RN so according to the above proposition there results the
equation SP + PB = BR + RN or what is the same SB: BN. Therefore point B
lies in the middle of the hypotenuse of the right angled triangle SON and conse-
quently equidistant from the points S, O and N. Hence it follows that OB = BN or
that the triangle OBN is an isosceles triangle. As PO: PB being half radii of the
same circle so is triangle POB also an isosceles triangle, and there result the following
equations -
=
=
If in this latter equation the 2 BON is inserted instead of POB there
2 BONBON
-
* PON
= 3 BON.
PON
a, b, c, h, (Figs. 4 to 9) are the four links of the rhombus shaped frame connected
together by studs d, e, f, g. n is the diagonal rod which passes through the studs
e, g, and is movable therein. The diagonal rod n also passes through and is movable
in a metal stud carrying a marking style p. The stud vis further pierced by
a guide rod which passes through a stud d and metal support f¹ and can be fixed
in these two parts by set screws as shewn. Stud fand support f¹ are each furnished
with a small spike by which they can be fixed upon the surface to be drawn upon.
x is a guide and fixing point on the stud d.
"
The different parts other than the metal parts mentioned may be of any suitable
material.
If now it is required to describe an ellipse of particular foci, the metal support f¹ is
fixed at one focus and the stud f at the other focus and so that the stud d lies outside
the focal distance. The rod is then fixed in the support and stud d by the set
screws, and a circle with the radius fd described with the point x of the stud d.
The style p will thereby describe an ellipse.
If it is required to describe a hyperbola of particular foci, the support fl and
the joint stud fare fixed in the respective foci so that the joint stud d lies between
the foci. The rod is then fixed in the support f¹ and stud d by the set screws and
a circle described with the point x. The style p will thereby describe the curve of a
byperbola.

page_text: To describe a parabola (Figs. 8, 9) the joint stud fis fixed in the araw
and the joint stud d drawn along a straight path or rail such for instance as S¹.
style will thereby describe a parabola.
The
lengthened
To divide the angle MOX (Fig. 3ª) into three equal parts, MO
past O and a circle of any convenient radius described from O. This circle cuts MO
at points P P¹. An angle of 45 is then drawn inwards from O cutting at F F¹ the
With P as a centre and radius P P¹ is described a circle
circle described from Ŏ.
and the instrument then set by placing the support f in F¹, the point f in F, and the
BOX is then the
style in P. The style is then moved until it cuts the circle described from P in R.
A line is drawn to connect R with P and this line halved at B.
third part of the angle MOX.
Having now particularly described and ascertained the nature of my said inven-
same is to be performed, I declare that what
tion and in what manner the
claim is-
An instrument for describing ellipses, hyperbolas, and parabolas and for trisecting
angles, consisting of links connected together to form a rhombus shaped frame; of a
rod n displaceably seated in two oppositely located link studs on which rod n the
metal stud carrying the marking style is mounted; and of a rod r passing through
said style stud and through the joint stud d, substantially as described and
0 shewn.
Dated this 26th dor of June 1899.
CAPITAINE & VON HERTLING,
15. Tooks Court, London, E.C., Agents for the Applicant.