A New or Improved Contrivance for Mechanically Determining Per Centages, and other Numerical Results.
Year: 1897

page_text: Date of Application, 19th May, 1897
Complete Specification Left, 19th Feb., 1898-Accepted, 19th May, 1898
A New or Improved Contrivance for Mechanically Determining
Per Centages, and other Numerical Results.
I, REDMUND NAISH, of 2, Proby Square, Blackrock, Dublin, Ireland, Civil Servant,
do hereby declare the nature of this invention to be as follows:-
The object of this invention is to accomplish by mechanical means the ascertain-
ment of per centages and other proportions and numerical results with equal accuracy
and greater simplicity, economy, and rapidity than by mechanical contrivances
hitherto in use so far as I am aware of.
In carrying out my invention I propose in the first instance to construct a simple
apparatus which for convenience I designate a "position finder," secondly I propose
to use this apparatus in combination with a scale of figures similar to that used in
some slide rules, namely, a scale of figures set forth in ordinary numerical order
upon a flat surface in parallel lines, the figures being separated from each other by
graduated linear spaces proportionate to the numerical intervals between the
logarithms of the natural numbers.
The "position finder" consists of a rectangular plate of wood or other suitable
material and of suitable dimensions.-One of the longitudinal edges of the plate is
chamfered so as to produce a sharp edge, and this chamfered surface is suitably
graduated and is marked by a certain fixed point which for convenience I designate
the "polar point." To this plate is connected by means of a swivel bracket mounted
on its upper surface, a moveable arm capable of free horizontal movement and
carrying a universal joint from which extends a second arm provided at its extremity
with a vertical pin or pointer called for convenience the "vectorial point." These
arms are of suitable length according to the size of the "position finder
and the dimensions of the scale to which the instrument is to be applied.-As thus
constructed the apparatus is used as in the following example :—
To find what per-centage of 952 is 684.
"
The "position finder is first applied to the scale so that the fixed "polar
point" will co-inside with the figure 952 and the chamfered edge with the line on
the scale.
The movable "vectorial point" is then brought to 684 on the scale.-Having
thus adjusted the movable or "polar point" in relation to the fixed or " vectorial
point "the instrument is lifted from the scale and reapplied so that the "polar
point" will co-inside with the figure 100 or 1 according to the deflection of the
vectorial point" and the chamfered edge lie along the line on scale whereupon
the movable or "vectorial point" will automatically indicate 71.8 which is the
required per-centage.
It is understood that there are three essental features contained in my
instrument, namely:-
(1) the "polar point" (2) the "vectorial point" (3) the line produced by the
chamfered edge.
Dated this 14th day of May 1897.
0 0011

page_text: A New or Improved Contrivance for Mechanically Determining
Percentages, and other Numerical Results.
I, REDMUND NAISH, of 2, Proby Square, Blackrock, Dublin, Ireland, Civil Servant, 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:
The object of this invention is to accomplish by mechanical means the ascertain-
ment of percentages and other proportions and numerical results with equal accuracy
and rapidity, and greater simplicity and economy than by mechanical contrivances
hitherto in use so far as I am aware; and, as an educational adjunct, to elicit some of
the properties of logarithms by graphic representation and a simple appliance adapted
for the tangible determination of position, involving an illustration of geometrical
and logarithmic theory combined, calculated to stimulate the interest of students.
In carrying out my invention I propose firstly to construct a suitable instrument,
and secondly, to use this instrument in combination with a scale similar to that used
in some slide rules:-namely, a scale of figures set forth in ordinary numerical
order on a flat surface in parallel lines,-the lines being equidistant one from the
other, and of equal length.
There is no necessary restriction as to the number of lines to be adopted. It will,
however, appear in the sequel that for purposes involving the ascertaining the
logarithm of a natural number, or vice versa, ten is the most convenient number of
lines, and after that, twenty. And as with the latter number the tabulation takes a
more compact shape, and suits the application of the instrument better, twenty is
the number adopted in this specification, in the diagram Fig. 1, and for each of the
component scales of diagram, Fig. 4.
The numbers shall consist of all the integral numbers in sequence from some one
power of 10 to the next higher, inclusive. Therefore they will be the integers from
1 to 10 both inclusive, or from 10 to 100, or from 100 to 1000, or from 1000 to 10,000,
&c. The scale from 100 to 1000, I have found to afford a good combination of
distinctness of result and compendiousness in size.
These numbers shall be in ordinary numerical sequence but at unequal intervals of
space. They may succeed each other from right to left on the top line, be continued
from right to left on the second line and so on. They shall be separated from each
other by graduated linear spaces proportionate to the mantissae of their logarithms.
For practical facility, this scale of numbers and graduations may be repeated so as
to form two scales in continuity one below the other, and it may also be repeated or
continued laterally so as to form four scales, as illustrated in Fig. 4, which arrange-
ment I designate for convenience a "combined table." In this diagram it will be
observed that the individual or single scales must be so joined laterally that every
two lines thus brought into junction and continuity shall show also a continuity of
the series of numbers marked upon these lines. Therefore, the first line of the single
scale which is at the right hand side shall be in continuity with the second line of
the single scale which is repeated at the left, and so on. In this manner any number
of single scales may be joined together.
For convenience I have adopted a combination of four single scales, as in Figure 4
aforesaid, and the examples hereinafter referred to shall be calculated in accordance
with this arrangement.
The total length of scale to be adopted varies according to the degree of accuracy
or distinctness to be obtained. For many ordinary purposes an aggregate linear

page_text: length of 80 inches for the lines of one scale is sufficient, affording results easily
distinct and accurate to 999 parts of 1000.
The direction of the sequence of the numbers from right to left mentioned above is
not essential.
The opposite direction might be adopted instead. Whichever is fixed upon,
however, should be maintained throughout all the lines.
Economy of space is served by expressing only every tenth one of the numbers,
and that with the terminal zero omitted.
The digits may furthermore be set sideways, e.g., 37 (which will stand for 370) may
be engraved thus
.
The instrument which constitutes the second part of my invention is illustrated in
Figs. 2 and 3, shewing a longitudinal side view, and a plan.
This instrument I designate for convenience the "position finder."
It consists of a rectangular plate A, of wood or other suitable material, one edge of
which is chamfered, and is marked by a certain fixed point which for convenience is
designated the "polar point," and which is indicated by the transverse arrow on the
chamfer in Fig. 3. The "polar point" must be in the chamfer edge but may be
selected therein at convenience.
From a swivel bracket C, swings horizontally an arm B, which carries by means of
a ball-and-socket joint a branch arm D, the extremity of which is provided with a
pointer E, which for convenience is designated the "vectorial point."
Standards H, H, are fixed on the upper surface for facilitating the movements of
the instrument.
The manner of using my invention is as in the following examples-sor
-some of which
explain how supplementary points marked upon the scale (Example 5) or supple-
mentary graduations marked upon the chamfered edge of the instrument
(Examples 6, 7, 8 and 9) serve in producing results, or in extending the application of
the invention.
It is understood throughout these examples whenever the "position finder" is
applied to the single scale or the "combined table" as the case may be, that the
chamfered edge is to lie along some line i.e. that it is not to make any angle with
the direction of the parallel lines on the scale or table.
Also, in these examples the first and last points of each single scale shall be
designated a "power point."
Thus, on the accompanying diagram, Fig. 4 there are nine power points the
position of each being indicated by a minute circle.

page_text: To divide 80850 by 462.
EXAMPLE 2.
Apply the polar point at 8085, and
Adjust the vectorial point to 462: then
Re-apply the vectorial point to a power point,
Whereupon the polar point will automatically indicate 175 the required
quotient.
To find the interest on £247 for 57 days at 5%.
(For the calculation of interest a subsidiary point may conveniently be marked
upon the combined table. For 5% rate select this point where the digits 137 2
determined by expressing as a decimal-would appear on a reversed scale if such
were set forth above the ordinary one. In Fig. 4 the subsidiary point for this
percentage is indicated by the letter S, in the margin. Upon the same principle
subsidiary points may be marked for other percentages.)
Apply the polar point at 247, and
Adjust the vectorial point to the subsidiary point, then
Re-apply the vectorial point at 570,
Whereupon the polar point will automatically indicate 1928 the digits of the
required interest £1.928.
3
To find the 11th root of 85650.
(For operations of this description involving the finding of the logarithm of a
number, and vice versa, a simple addition or modification is requisite both to the
combined table and to the instrument.
(1) The lines of the combined table are to be numbered at the right hand side
thus: If the single scale be set forth in ten lines they are each respectively, to be
numbered in succession at the side 0. 1. 2. &c., on to 9 commencing from the top
line which is to be zero, but if the scale be set forth in twenty lines as in the single
scale Fig. 1 and the combined table Fig. 4 each second one is to be numbered as
before. 0. 1. 2. &c., on to 9.
The use to be made of these numbers shall be presently explained. Their meaning
may be taken to be 1°, 10, 10, &c. of the linear length of the total scale from the
commencing point.

page_text: (2) The instrument is to be furnished with a series of graduations upon the
chamfered edge each graduation representing the hundredth part of the tenth part of
the total length of the scale, and these graduations should be numbered consecutively
beginning at zero at the left and running as indicated by the arrow in Fig. 3.
One hundred graduations are to be marked if the scale be set forth in ten lines:
but if it be set forth in twenty lines, as illustrated in Figs. 1, and 4, then fifty
graduations only are to be marked.
If great accuracy be required, and if the length of the scale admits, these
graduations may be further subdivided decimally).
Apply zero of the graduated edge at 8565. The edge will then meet the boundary
line where the scale is numbered 9 at the side and the boundary line will meet the
edge at graduation 32.7. The number 9327 is thus obtained, and it is the mantissa
of the original number 85650. To this mantissa prefix 4 for characteristic, giving
logarithm 4-9327. Divide this by 11 which may be done on the combined table in
the manner set forth in Ex. 2-or by common arithmetic. The result is 0.4484
and this is the logarithm of the required root. To convert it into a natural number
pursue the converse method to the above, i.e. bring the graduated edge to the line
marked 4 at the side adjusting graduation 48.4 to the boundary line-whereupon
zero of the graduated edge will automatically indicate 2808 the digits of the
required root 2.808.

page_text: Take the mantissa of 105, viz.: 02119 (if the condition were 6 per cent., we should
take instead the mantissa of 106). This may be found on the combined table as
shown in foregoing example or, preferably, it may be recorded upon some convenient
blank space on the margin of the table. Even were a number of mantissas
corresponding to the several percentages in frequent use, thus recorded, it would
not, it is believed, inconveniently load the table with figures.
Subtract 1
Multiply the mantissa by 27 producing 0.57213. Regard this as a logarithm,
and as shown in Examples 6 and 8 ascertain its natural number 3.735.
from unit figure leaving 2-735.
Apply the polar point at 2735, and
Adjust the vectorical point to 5 in the table above (rate per cent.):
Re-apply the vectorial point at 3333 (since £3 6s. 8d. £3.333).
Whereupon the polar point will automatically indicate 1824 the digits of the
required result £182 4s.
More examples might be set forth; but it is believed that enough have been
given to demonstrate the varied usefulness of the invention.
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
First The use in combination with a logarithmic scale as described and
illustrated in Fig. 1 or a "combined table" as described and illustrated in Fig. 4,
of an instrument the essential parts of which are (1) a line; that is the line along
the chamfered edge of the instrument, (2) a fixed point in the line; that is the point
marked by the transverse arrow in Fig. 3, (3) a moveable point outside of the line,
that is the point carried by the moveable arm of the instrument, whereby the
distance between any two points on the scale or table and the direction of their
joining line relatively to the direction of the parallel lines are fixed by the
application of the instrument; and by the removal and re-application of which latter
to another position, one of its points may be made to coincide with any third point
on the scale or table, whereupon the second point of the instrument will
automatically indicate a fourth point of the logarithmic scale, or table, the natural
number corresponding to which shall be a fourth proportional to the three natural
numbers corresponding to the three previous points on the scale, or table.
Second: The use and application for the purposes of arithmetical calculations
of an instrument constructed, arranged and operating substantially as herein
described and illustrated in Figs. 2 and 3 of accompanying drawings.
1000
J. K. FAHIE & SON,
Agents for Applicant, 9, Westland Row, Dublin.
J. K. FAHIE & SON,
Agents for Applicant, 9, Westland Row, Dublin.