A NEW OR IMPROVED INSTRUMENT FOR MECHANICAL DRAWING AND MEASURING
Year: 1873

page_text: PROVISIONAL SPECIFICATION left by Johann Ernst Friedrich Lüdeke
at the Office of the Commissioners of Patents, with his Petition,
on the 3rd September 1873.
I, JOHANN ERNST FRIEDRICH LUDEKE, of 15, Wilmot Place, Camden
Town, in the County of Middlesex, do hereby declare the nature of the
said Invention for "A NEW OR IMPROVED INSTRUMENT FOR MECHANICAL
DRAWING AND MEASURING," to be as follows:-
My Invention consists of a new or improved instrument whereby the
relative proportions of the square and circle may be determined and
drawn to an accurate scale, other problems appertaining thereto being
also solved by its use.
In order to explain the theory upon which the construction of the
instrument is based, I annex the accompanying diagrams, in which,
Sheet 1, Fig. 1, represents three circles, the outer one described with the
full radius; the inner circle is described with any required radius, and
the intermediate circle is equidistant from or between the other two.
Now if the circles are divided into any number of parts, say 12, by lines
radiating from the centre the area or cubical contents thereof shall

page_text: be equal to the parallelogram shewn divided into 12 equal parts,
equal to the distance between the points of intersection or radii of the
intermediate circle, for inasmuch as the periphery of the outer arc
exceeds the length of the parallelogram by the same distance as the
internal arc is shorter, it must follow that the intermediate arc will be t
the true length of the required parallelogram, the width of which must
be equal to the length of the intersecting radii.
In Fig. 2 two circles are described, the inner one with half the radius
of the outer; the diameter of the outer circle is divided into 28 exactly
equal parts, 18 of which are shewn on a lower line or scale drawn as a 1
tangent to the inner circle and parallel with the diameter; the lower half
of the inner circle is divided into 18 equal parts; and its periphery is
exactly equal to the 18 divisions of the scale which are intended to form
the short sides of a parallelogram whose longer sides are equal to the
full diameter of the circle.
]
If the lower half of the inner circle were a disc of metal and were
rolled over the lower scale of measures, its periphery would exactly
equal it, and by reference to Fig. 3, Sheet 2, I shew the full circle
divided into two portions arranged so that their peripheries are in
contact, their respective diameters being bisected at the points a, a, a
equal to the length of the periphery of the internal semicircle, or 18
parts out of 23 of the full diameter, here shown by the lines b, b. From
this example it will follow that if a line of any length be divided into 23
equal parts and 5 parts be removed the remaining 18 parts shall form
the short sides of a parallelogram whose longer sides shall be equal to
the whole 23 parts, this figure being reduced to a square will then
equal the area of a circle whose diameter is equal to that of the longer
side of the parallelogram. The parallelogram thus formed is shown at
Fig. 4 with the measurements of the respective sides and the area
contained thereby.
Fig. 5 shows the true square with its area equal to that of Fig. 4,
and consequently the true side of a square is formed equal to the area
of the parallelogram, Fig. 4, and of the circle shewn in Figs. 2 and 3,
and by reference to the latter Figure the parts c, c, c, c, separated from
the two semicircles by the lines b, b, are equal to the parts d, d, con-
tained by those lines and the arcs of the semicircles. Upon these
principles the instrument is constructed; a simple example of pro-
portional dividing compasses is shewn at Fig. 6, Sheet 2, but of course

page_text: drawing compasses may be made on the same principle, or the dividers
may have suitable pen legs for Drawing. The side of the square, and
the corresponding diameter of the circle being proved by the diagrams
or corresponding calculations according to the rule laid down, the size
of instrument is determined, and a four sided figure being formed
accordingly from the four opened extremities or points of the instrument
proposed, diagonals are drawn the intersection of which serves as the
point or centre for the joint of the dividers or compass, which will
then determine the relative proportions of the circle and square, as
well as other problems referring or relating thereto.
The lower or longest legs of the instrument indicate the diameter
of the circle and the upper or shortest legs shew the corresponding
side of the square required.