AN IMPROVED CALCULATING INSTRUMENT."
Year: 1881

page_text: LETTERS PATENT to James Baldwin Fearnley, of Castleford, in the County
of York, Consulting Engineer, for an Invention of "AN IMPROVED
CALCULATING INSTRUMENT."
PROVISIONAL SPECIFICATION left by the said James Baldwin Fearnley at
the Office of the Commissioners of Patents on the 1st March 1881.
JAMES BALDWIN FEARNLEY, of Castleford, in the County of York, Consulting
Engineer. "AN IMPROVED CALCULATING INSTRUMENT."
The Invention relates to that class of calculating instrument in which a movable
logarithmic scale is employed, in combination with fixed and moveable indices or
pointers.
In instruments of this kind, like that known as the slide rule, the divisions
towards one end of the logarithmic scale are necessarily smaller than those towards
the other end, and consequently the value of those divisions is less easily and less
accurately read off than that of the larger divisions, and the accurate adjustment of
the indices to them is also more difficult.
In this Invention the divisions at what may be called the upper end of the
logarithmic scale are made larger in proportion than those on the corresponding
part of an ordinary logarithmic scale, and are therefore more easily read off and
adjusted to the indices, or the indices to them. In order to effect this I employ a
dial of metal, wood, paper, or other suitable material, on the surface of which is
described a diverging spiral line of any required number of convolutions, each of
which is divided into the logarithmic scale in such manner that the commencement
and end of each such scale is on one radial line. The dial is capable of revolving round
a centre fixed to a suitable base plate. A wire, thread, or other suitable index
extends radially over the dial from the centre thereof to the circumference, being
fixed to the centre or to the base plate, or to both. Another or several other
similar radial index or indices is or are adapted to turn on the centre, so as to
sweep over the whole surface of the dial occupied by the before mentioned
logarithmic scale. Each of the aforesaid indices may be furnished with an

page_text: adjustable pointer sliding longitudinally thereon. On the outer edge of the dial is
described a circle divided into ten equal parts, with as many subdivisions as may be
required.
By manipulating the dial and index or indices in certain ways the various
arithmetical operations of multiplication, division, evolution and involution, and so
forth may be performed with extreme accuracy.
It will be readily understood that as each successive convolution of the spiral,
reckoning outwards from the centre of the dial, is larger than the preceding one,
the logarithmic scale marked on it will have its divisions larger than those of the
said preceding convolution in the same proportion as the radii of the respective
convolutions on which such divisions are found, that the linear divisions on the
spirals represent the natural numbers, and the numbers of the convolutions counted
from the centre indicate the corresponding mantissœe of the logarithms of those
numbers, the other figures of the logarithms being shown on the outer decimal circle
of the dial.
The dial may be made flat, conical, as a spherical segment, or in the form of any
other solid of revolution in which the generating radius increases as the base of the
solid is approached.

page_text: SPECIFICATION in pursuance of the conditions of the Letters Patent filed
by the said James Baldwin Fearnley in the Great Seal Patent Office on the
1st September 1881.
JAMES BALDWIN FEARNLEY, of Castleford, in the County of York, Consulting
Engineer. "AN IMPROVED CALCULATING INSTRUMENT."
The Invention relates to that class of calculating instrument in which a moveable
logarithmic scale is employed in combination with fixed and moveable indices
or pointers.
In instruments of this kind, like that known as the slide rule, the divisions
towards one end of the logarithmic scale are necessarily smaller than those towards
the other end, and consequently the value of those divisions is less easily and less
accurately read off than that of the larger divisions, and the accurate adjustment of
the indices to them is also more difficult.
In this Invention the divisions at what may be called the upper end of the
logarithmic scale are made larger in proportion than those on the corresponding
part of an ordinary logarithmic scale, and are therefore more easily read off and
adjusted to the indices, or the indices to them. In order to effect this I employ a
dial of metal, wood, paper, or other suitable material, on the surface of which is
described a diverging spiral line of any required number of convolutions, each of
which is divided into the logarithmic scale in such manner that the commencement
and end of each such scale is on one radial line. The dial is capable of revolving
round a centre fixed to a suitable base plate. A wire, thread, or other suitable
index extends radially over the dial from the centre thereof to the circumference,
being fixed to the centre or to the base plate, or to both. Another or several
other similar radial index or indices is or are adapted to turn on the centre, so as to
sweep over the whole surface of the dial occupied by the beforementioned logarithmic
scale. Each of the aforesaid indices may be furnished with an adjustable pointer
sliding longitudinally thereon. On the outer edge of the dial is described a
circle divided into ten equal parts, with as many subdivisions as may be
required.
By manipulating the dial and index or indices in certain ways the various
arithmetical operations of multiplication, division, evolution and involution, and so
forth may be performed with extreme accuracy.
It will be readily understood that as each successive convolution of the spiral,
reckoning outwards from the centre of the dial, is larger than the preceding one,
the logarithmic scale marked on it will have its divisions larger than those of the
said preceding convolution in the same proportion as the radii of the respective
convolutions on which such divisions are found, that the linear divisions on the
spirals represent the natural numbers, and the numbers of the convolutions counted
from the centre indicate the corresponding mantisso of the logarithms of those
numbers, the other figures of the logarithms being shown on the outer decimal circle
of the dial.
The dial may be made flat, conical, as a spherical segment, or in the form of any
other solid of revolution, in which the generating radius increases as the base of the
solid is approached.
:
Figure 1 of the Drawings represents an apparatus constructed in accordance with
my Invention; Figure 2 is a section of the same.
A is a base plate constructed of wood, metal, or other suitable material; B is a
circular plate, made preferably of metal, and adapted to revolve round a centre
pin C fixed in the base plate A. Round the outer edge of the plate B is fixed a
rim D, which may be conveniently constructed of a bent metallic tube, with a slit E

page_text: formed round the outer edge thereof. G is a circular dial of metal, wood, card-
board, paper, or other suitable material, also adapted to turn freely on the centre
pin C, and having described thereon a diverging spiral line or volute, which is
divided into a logarithmic scale in such manner that the commencement and end of
such scale is on the radial line R. In the example shown there are two such
scales, namely, an inner one from 100 to 1000, and an outer one from 1000 to
10,000 but it will be understood that the number of scales on the same dial may be
varied at will; for instance, the inner scale from 100 to 1000 may be dispensed
with, and the outer one from 1000 to 10,000 only employed; or other similar
scales may be added at pleasure by enlarging the diameter of the dial G. F is a
wire or thread fixed at one end to a ring capable of turning on the centre pin C,
and extending radially over the dial G is secured at the other end to the rim D of
the plate B; f is a bead or other index or pointer sliding on the wire or thread F;
M is a similar wire or thread fixed at one end to another ring capable of turning
on the centre pin C, and extending radially over the dial G is secured at the other
end to a carriage H, adapted to slide round the rim D, being guided thereon by a
projection taking into the slit E formed therein; m is a bead or other index or
pointer sliding on the wire or thread M. The outer edge of the dial G is or may
be divided into ten equal parts, with as many subdivisions as may be required. It
will now be understood that, as before stated, the linear divisions on the spirals or
volutes represent the natural numbers, and the numbers of the convolutions counted
from the commencement of each logarithmic scale indicates the first figure of the
mantisse of the logarithms of those numbers, the other figures of the logarithm being
shown on the outer decimal circle of the dial at the point thereof which occurs on
an extension of the radial line passing from the centre through the said natural
number; for example, taking the natural number 805, and placing the moveable
wire or thread M over it, or (what amounts to the same thing) turning the dial G
round until the number 805 comes exactly under the fixed wire or thread F, or the
moveable one M, it will be found that the said number 805 occurs on the ninth
diverging spiral or volute counted from the beginning of the scale, and that the
wire or thread F or M cuts the decimal scale at the circumference of the dial Gat
the fifty eighth division nearly from zero, showing the mantissœe of the logarithm of
that number to be 9058 nearly.
The apparatus is used in the following manner :—
For Multiplication. Place the radial line R under the fixed wire or thread F,
and the index f over 100 on the spiral scale. Place the moveable wire or thread M
in the radial position, crossing the multiplier on the spiral scale, and the index m
immediately over such multiplier; then bring the multiplicand on the spiral scale
under the fixed wire or thread F, and the product will be found under M at the
.same distance from m that the multiplicand is from f.
For Division. Place the divisor on the spiral scale under the fixed wire or thread F,
and index f over the said divisor. Place the moveable wire or thread M in the
radial position, crossing the dividend on the spiral scale, and the index m imme-
diately over such dividend; then bring the radial line R under the fixed wire or
thread F, and the quotient will be found under M at the same distance from m that
100 is from f.
For Involution and Evolution. Up to the third power it is best to proceed as in
multiplication, but for higher powers and for roots by logarithms thus:-Find the
mantissa of the logarithm of the number whose power or root is required by
bringing the said number on the spiral scale under either of the wires or threads F
or M; the number of the spiral or volute on which the number occurs, counted
from the beginning of its logarithmic scale, will show the first figure of the
mantissa, and the other figures can be read off on the circumferential decimal scale
at the point where the wire or thread F or M crosses it. Add to the mantissa or
decimal part of the logarithm thus found the logarithmic index or integer figure
corresponding to the number, and if the power is required multiply the whole
logarithm so obtained by the index of the power required, which will give the

page_text: logarithm of the power of the number, from which the corresponding natural number
is then found on the instrument by the reverse of the process for finding the
logarithm of a number. If the root be required instead of the power, the whole
logarithm is to be divided by index of the root required, whereby the logarithm
of the root is obtained, from which the corresponding natural number can be found
on the instrument, as before.
For Proportion. Place the first term (the divisor) under f, m, over either of the
other terms, then bring the remaining term under F, and the fourth term is found
at the same distance from m that the other term is from f. The plate B is inter-
posed between the base plate A and the dial G to enable the instrument to be
turned easily into the position best suited for the operation without the incon-
venience of having to turn the base plate as well. If preferred the plate B may be
dispensed with, the rim D being then fixed to the base plate A; or the base plate A
may be dispensed with, and the plate B retained, in which case the centre pin C
must be fixed to the plate B.
Figure 3 shows a modification of the dial, in which the numbers on the inner
logarithmic scale formed on five spirals or volutes indicate the squares of the
numbers on the same radial lines comprising the outer logarithmic scale formed on
ten spirals or volutes, consisting of two sets of five each; for example, let the
number 90 on the spiral or volute of the outer series marked 4 indicate 9.0, its
square 81 is shown on the spiral or volute marked 4 of the inner series at the
point where the radial line crosses it. Similarly the number 20 on the spiral of
the second series marked 2 has its square 400 on the spiral of the inner series at
the point where the radial line crosses the same, and so on for other numbers.
If the inner scale be made loose so as to turn in the outer one the side of a square
equal in area to a parallelogram can be found, as on the ordinary slide rule. Fixed
and moveable indices may be employed in connection with this dial, as described,
in reference to the dial shown in Figures 1 and 2.
Scales showing the decimal parts of a pound sterling, a ton weight, a mile, a
yard, a year, and so forth, may be applied, as shown, near the edges of the
base plate A, for the purpose of facilitating the use of decimal equivalents when
employing the instruments in calculations involving qualities of those kinds.
Having now described the nature of the said Invention, and the manner in
which the same is to or may be carried into effect, I would have it understood that
what I claim is,―
Firstly. The improved calculating instrument, consisting of a moveable dial, on
which is described a diverging spiral line divided into one or more logarithmic
scales, combined with fixed and moveable indices or pointers, and with or without
a decimal scale at the circumference of the said dial, substantially as hereinbefore
described and illustrated in the Drawings.
Secondly. In calculating instruments of the kind herein described the employ-
ment of moveable dials with diverging spiral lines described thereon, and divided
into logarithmic scales, whether such dials be flat, conical, or in the form of any
other solid of revolution in which the generating radius increases as the base of the
solid is approached.
In witness whereof, I, the said James Baldwin Fearnley, have hereunto set
my hand and seal, this Thirtieth day of August, in the year of our Lord
One thousand eight hundred and eighty one.
J. B. FEARNLEY. (L.S.)