Improvements in Calculating Rules and the like.
Year: 1899

page_text: Improvements in Calculating Rules and the like.
I, WILLIAM JOHN GOUDIE, Engineer, of 92, Albert Drive, Crosshill, Glasgow,
in the County of Lanark, do hereby declare the nature of this invention to be
as follows:-
My invention relates to calculating instruments such as slide rules, and has
for its object to enable the calculations required in the designing or adjusting
the proportions of compound or multiple-expansion engines to be rapidly and
accurately made, so that many calculations may be made on different assump-
tions, till the desired results be obtained.
My invention consists in an improved form of calculating rule, adapted to this
purpose, and in improved arrangements and combinations of scales.
My improved calculating rule has two main parts, which may each be used
separately or may be combined in one rule, each forming one face of the rule.
One of these parts I call the "synthetic," and the other I call the "analytic"
part each part is complete in itself.
The synthetic part consists essentially of a combination of scales each repre-
senting a series of values of one of the various factors from which the total
indicated horse power of a multiple expansion engine can be computed, arranged
and adapted to be moved relatively so that different values of the factors can be
chosen at will and dealt with, and the resulting horsepower, for the chosen
cylinder proportions, read off.
The analytic part of the rule consists essentially of a combination of scales,
each representing a series of values of one of the indicated horsepower factors
or its constituent elements, or the constituent elements of the factors of cylinder
proportions, arranged and adapted to be relatively moved so that the more
complex factors can be separated into their constituent elements; and by taking
different values for the factors themselves the necessary elements can be deter-
mined and an analysis of the conditions made, for any chosen distribution of
the total power in the various cylinders.
Constructionally the two parts of the rule are alike, the only difference being
in the scales placed on each.
The synthetic part consists of a fixed rule carrying two sliders; these slide
against one another, edge to edge, at the centre of the rule and are provided with
suitable scales as described below. The outside edges of these sliders are pro-
vided with certain scales, and they move edge to edge with and read against
suitable scales provided on the fixed parts of the rule. The analytic part of the
rule is of the same construction, but its scales are different. The scales on
both the synthetic and the analytic parts are partly logarithmic and partly
equal division scales.

page_text: At certain places on the rule, both on the synthetic and the analytic paris
index-arrow or gauge lines are provided, and these read against the scales placed
on the other parts of the rule moving in relation to the first parts, or vice versa.
I may, without departing from my invention, arrange and combine the scales
on one or more discs or rings having motion in relation to one another and to
suitable gauge points or index-arrow lines, or on one or more suitable cylinders.
In carrying my invention into effect, according to one modification 1 provide
on the synthetic face of the rule, on the left hand part the following scales and
index-arrow lines, a logarithmic scale for low pressure cylinder diametres (which
can also be used for intermediate-pressure cylinder diameters); a logarithmic
scale for cylinder ratios; an equal division scale for back pressures, and an index-
arrow line (called hereafter arrow E.")
..
....
On the left hand slider and along its left hand edge (sliding edge to edge
with the above mentioned scales), a logarithmic scale for high pressure cylinder
diametres; an index arrow line (called hereafter arrow A.") and an equal
division scale (called hereafter scale 2.") Along the right hand edge of the
slider an index arrow line (called hereafter "arrow B); a logarithmic scale for
mean pressures (called hereafter the upper mean pressure scale"); a logarith-
mic scale for number of expansions (called hereafter the lower expansion
scale," and an index arrow line (called hereafter arrow D.).
On the right hand slider and along its left hand edge (sliding edge to edge
with the scales on the right hand edge of the other slider) a logarithmic scale for
cut-off in the cylinder; an index-arrow line (called hereafter arrow G") and a
logarithmic scale for initial pressures.
Along the right hand edge of the slider an index-arrow line (called hereafter
arrow "); a logarithmic scale for piston speeds, and a logarithmic scale for
mean pressures (called hereafter the lower mean-pressure scale").
"
These scales slide edge to edge with the scales on the right hand fixed part,
which are, a logarithmic scale for number of expansions (called hereafter the
upper expansion scale"); a logarithmic scale for indicated horse power; a
logarithmic scale for diagram or efficiency factors, and an index arrow line
(called hereafter arrow F.)
.
I will now give a few examples of the various operations, and the modes of
performing these, to illustrate some of the uses of this synthetic part of my
calculating rule.
(1) To find the cylinder ratio place the high-pressure-cylinder diameter
against the low pressure cylinder diameter and read the total cylinder ratio
opposite arrow A."
These two scales can be used, also, to find the ratios for the intermediate
cylinders. To find the necessary diameters to give a certain chosen cylinder
ratio, place the "arrow A" against the chosen ratio and any pair of diameters
which come opposite one another on the high pressure and low pressure diameter
scales will give the ratio required.
"
(2) To find the nominal number of expansions for a chosen cut off, place the
arrow Aagainst the cylinder ratio: place the cut off against the " arrow B,"
and read the number of expansions on the " upper expansion scale" opposite
the arrow C."
arrow A
To find the cut off to give a certain number of expansions, place the "
against the cylinder ratio; place the "arrow ("against the number of expan-
sions on the " upper expansion scale," and read the cut off opposite arrow B."
(3) To find the mean effective pressure in any evlinder, or the mean effective
pressure referred to the low pressure cylinder, place the " arrow against
the initial pressure in the cylinder in question (in the case of the referred mean
pressure this is the initial pressure in the high pressure cylinder) on the initial
pressure scale and read result I on this scale opposite the number of expansions,
(lower expansion scale"); place the back pressure against this result 1 on

page_text: "
ii
scale 2" and read result 2 opposite the arrow place the arrow F"
against this result 2 on the Tower mean pressure scale," and opposite the
chosen diagram factor read the mean effective pressure required.
By an inversion of this series of operations under (3), the number of expansions
corresponding to any given mean pressure can be determined, and the necessary
cut off then found by operation (2).
(4) To find the total indicated horsepower, place the high pressure diameter 10
against the chosen low pressure diameter; place the arrow G against the
referred mean pressure (determined by operation (3) or otherwise) on the
upper mean pressure seale," and opposite the chosen piston speed read the
indicated horse power. To find the necessary diameter of low pressure cylinder
for a given horsepower (the necessary referred mean pressure having been
determined from the number of expansions to be allowed) place the chosen piston
speed against the indicated horse power; place the referred mean pressure (on
the upper mean pressure scale") opposite the "arrow G," and opposite the
high pressure diameter 10 read the low pressure diameter required.
(5) For the case of a triple expansion engine with two low pressure cylinders,
to determine the diameters of these cylinders equivalent to the single low pres-
sure cylinder, place the high pressure diameter 14 against the single low pressure
cylinder diameter and read the equivalent diameter of cylinder, for the double
low-pressure cylinder engine, opposite the high pressure diameter 10.
"
I provide, on the analytic face of the rule, on the left hand fixed part, a log-
arithmic scale (called hereafter pressure scale 7"); a logarithmic scale for
diagram factors, and an index-arrow line (called hereafter arrow H.")
"
"
On the left hand slider, and along its left-hand edge (sliding edge to edge
with the above mentioned scales) a logarithmic scale (called hereafter pressure
scale 6,) and a logarithmic scale for mean pressures. Along the right hand
edge of the slider an index arrow line (called hereafter arrow. L,") and an
equal division scale (called hereafter pressure "scale 3.")
On the right hand slider, and along its left hand edge (sliding edge to edge
with the scale on the right-hand edge of the other slider) an equal division
scale (called hereafter " pressure and temperature scale 4.")
Along the right hand edge of the slider an index arrow line (called hereafter
arrow N) a logarithmic scale for number of expansions; an index arrow
line (called hereafter arrow K,") and a logarithmic scale (called hereafter
pressure scale 5.")
"
".
These scales slide edge to edge with the scales on the right hand fixed part,
which are, a logarithmic scale for initial pressures: a logarithmic scale for cylin-
der ratios, and an index-arrow line (called hereafter arrow M.")
I will now give a few examples of the various operations, and the modes of
performing these, to illustrate some of the uses of this analytic part of my
calculating rate.
(1) From the initial pressure, the mean effective pressure, and the number of
expansions in any cylinder to find the initial pressure in the next cylinder,
place the suitable diagram factor against the mean effective pressure, and
read result 3 opposite arrow place "arrow K" against the initial pressure
and read result 4 opposite the number of expansions: place result 3 on
pressure
scale 3" against result 4 on pressure scale 4," and read the required initial pres-
sure on pressure scale 4" opposite arrow L."
**
"
(2) To find the back pressure or mean back pressure in this cylinder, proceed
as in operation (1) using a different value of diagram factor.
(3) From the initial pressure. the mean back pressure, and the
mean effective pressure to find the number of expansions in the
cylinder, place the diagram factor (suitable for back pressure in operation
(2)) against the mean effective pressure, and read result 3 opposite
H"; place arrow L" against the mean back pressure on
"
1
"
arrow
pressure scale 4,"
and opposite result 3 on pressure scale 3" read result 5 on pressure scale 4";

page_text: place arrow K" against the initial pressure and opposite this result 5 on the
initial pressure scale read the number of expansions.
The cut off to give this number of expansions can then be determined by the
synthetic part of the rule.
"
(4) To find the initial load on the piston determine the initial and back pres-
sures by operations (1) and (2), then place the back
pressure on
pressure scale 3
against the initial pressure on pressure scale 4," and opposite arrow Lon
pressure scale 4" read result 5 which is the effective initial load on the piston
in pounds per square inch.
"
To express this as an equivalent pressure per square inch of low pressure
piston area, place the ratio of this cylinder to the low pressure cylinder, against
result 5 on pressure scale 5, and read the initial load, in pounds per square inch,
referred to the low pressure piston, opposite
arrow M."
"
arrow N
(5) To find the range of temperature in any cylinder place the
against initial pressure 10; place the proper back pressure, on pressure scale 6,"
against the initial pressure on pressure scale 7," and opposite the arrow L,"
on pressure and temperature scale 4," read the range of temperature in degrees
Fahrenheit.
MARKS & CLERK,
18. Southampton Buildings, London, W.C.,
13, Temple Street, Birmingham, and
25, Cross Street, Manchester, Agents.
I, WILLIAM JOHN GOUDIE, of 92, Albert Drive, Crosshill, Glasgow, Engineer,
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:-
My invention relates to calculating instruments such as slide rules, and its
object is to enable the calculations required, when designing or adjusting the
proportions of multiple expansion engines, to be rapidly and accurately per-
formed so that many calculations may be made on different assumptions till
the desired results are obtained.
My invention consists in an improved form of calculating rule adapted to
this purpose, and in improved arrangements and combinations of scales.
This improved calculating rule has two main parts: each part may be used
separately or the two parts may be combined in one rule, each forming one
face of the rule or otherwise.
One of these parts I call the
each part is complete in itself.
The synthetic part consists essentially of a combination of scales, each
representing a series of values of one of the various factors from which the
total indicated horse of a multiple expansion engine can be computed (without
direct reference to the particular distribution of this total power in the various
cylinders), the scales being arranged and adapted to be moved relatively so
that different values of the various factors can be chosen at will and dealt with,
and the resulting horse power, for the chosen cylinder proportions read off:
or, conversely, so that the suitable cylinder proportions for the chosen indicated
horse power can be determined.
synthetic" and the other the "analytic" part;

page_text: The analytic part of the rule consists essentially of a combination of scales
each representing a series of values of one of the indicated horse-power factors,
or its constituent elements, or the constituent elements of the factors of cylinder
proportions, the scales being arranged and adapted to be relatively moved so
that the more complex factors can be separated into their constituent elements;
and, by taking different values for the factors themselves, the necessary elements
can be determined, and an analysis of the conditions made for any chosen dis-
tribution of the total power in the various cylinders.
Constructionally the two parts of the rule are alike, the only difference being
in the scales placed on each.
According to one modification of my invention the synthetic part is formed
with a fixed double rule carrying two sliders: the analytic part is of the same
construction but its scales are different.
The scales on both the synthetic and analytic parts are partly logarithmic and
partly equal division scales.
At certain places on the rule, both on the synthetic and on the analytic part,
index arrow or gauge lines are provided; and these read against the scales
placed opposite and moving relatively to them. Without departing from my
invention I may also arrange and combine my scales on suitable discs or rings
or cylinders, the scales having motion relatively to each other and to suitable
index arrow or gauge lines.
Referring now to the accompanying drawings which illustrate my invention:-
Figure 1 shows the synthetic and Figure 2 the analytic face of the rule; but I
wish it to be understood that these figures are merely diagrammatic.
Referring in the first place to Figure 1, A is the left hand fixed part and A²
is the right hand fixed part of the rule, while S is the left hand slider and S
the right hand slider.
On the fixed part A are provided a logarithmic scale for low pressure
cylinder diameters; a logarithmic scale b for cylinder ratios; an equal division
scale c. for back pressures and an index arrow or gauge line B.
On the left hand edge of the slider S₁ I provide a logarithmic scaled for high
pressure cylinder diameters, an index arrow line E, and an equal division scale c.
On the right hand edge of the slider S', I provide an index arrow line G; a
logarithmic scale f, for mean pressures, a logarithmic scale g for number of
expansions and an index arrow line A.
Un the left hand edge of slider S2, I provide a logarithmic scale h, for cut
off, in the cylinder; an index arrow line 1) and a logarithmic scale k for initial
pressures.
an
On the right hand edge of slider S2, are situated an index arrow line F,
index arrow line K, a logarithmic scale i for stroke of piston, a logarithmic
scale for piston speed and a logarithmic scale m for mean pressures.
On the fixed part A2, I provide a logarithmic scale for number of expansions,
a logarithmic scale o, for indicated horse power, a logarithmic scale p, for
diagram or efficiency factors and an index arrow line C.
To illustrate the application of this invention to the provisional methods of
designing and adjusting cylinder proportions of multiple expansion engines, I
shall now describe how the principle operations and their inversions are per-
formed, by means of the synthetic part of the rule.
1. To determine the means effective pressure referred to the low pressure
cylinder, place the index arrow line A against the initial pressure, on scale k,
and, opposite the nominal number of expansions on scale g. read result on
scale k. Place this result 1 on scale e against the back pressure in the low pres-
sure cylinder on scale e, and, opposite the index arrow line B, read result 2 on
scale e.
Place result 2 on scale m, against the index arrow line C, and opposite
the diagram factor in scale, read the referred mean pressure on scale m.
Similarly, if any other of these five factors is unknown, and the other four

page_text: known, it can be determined by adjusting the known values in the relative
positions as explained above.
2. To determine the diameter of the low pressure cylinder.
Place the piston speed, on scale, 7, against the total indicated horse power on
scale place the referred mean-pressure, on scale f, against the-index arrow
line Dand, opposite the high pressure diameter 10, (marked with two arrow
heads), on scale d, read the diameter of low pressure cylinder on scale a.
Similarly, if any other of these four factors is unknown and the three others
known, the unknown factor can be determined by adjusting the slider so that
the known quantities are brought into their proper relative positions, when the
unknown factor will be found in its relative position opposite one of these.
For the case of a triple expansion engine with two low pressure cylinders, in
order to find the diameter of each of the smaller cylinders, place the high pres-
sure diameter 14.1 on scale (marked with one arrow head) against the single
low pressure diameter (as determined above) on scale a and, opposite the high
pressure diameter 10 (marked with two arrow heads) read the diameter of the
smaller cylinder on scale #.
3. To determine the diameter of the high pressure cylinder.
Place the index arrow line E against the total cylinder ratio, on scale b, and,
opposite the low pressure cylinder diameter, on scale a, read the diameter of the
high pressure cylinder on scale d.
Similarly, if the cylinder diameters are known, by placing their values on
scales and against each other, their ratio can be found opposite the index
arrow line E, on scale b.
In the same manner the diameter of the intermediate pressure cylinder or
cylinders can be found from scales b, and d.
4. To determine the cut off in the high pressure cylinder.
Place the index arrow line E against the total cylinder ratio, on scale place
the index arrow line F against the nominal number of expansions, on scale a
and, opposite the index arrow line G, read the cut off in the high pressure
cylinder, on scale h.
Similarly the number of expansions can be determined by placing the known
cut off on scale against the index arrow line G and reading the number of
expansions opposite the index arrow line F, on scale n.
Referring now to Figure 2, which illustrates the analytic face of the rule,
13 is the left hand and B2 the right hand fixed part; while T and T2 are
respectively the left hand and right hand sliders.
On the fixed part B are provided a logarithmic scale y a logarithmic scale r,
and an index arrow line I. On the left hand edge of T' are located a logarithmic
scales and a logarithmic scale (:
On the right hand edge of T I provide an index arrow line L and an equal
division scale u.
On the left hand edge of T2 is provided an equal division scale r.
On the right hand edge of T2 I provide an index arrow line M, a logarithmic
scale w, and index arrow line N and a logarithmic scale r.
On the fixed part 13 are situated a logarithmic scale y, a logarithmic scale =
and an index arrow line 0.
To illustrate the application of this invention to an analysis of the conditions
of temperature and pressure ranges in the different cylinders of multiple ex-
pansion engines, initial loads etc., and to the adjustment of cut-offs to attain
proper distribution of the total power in the various cylinders, I shall now
described how the principal operations, to effect these objects, are performed by
means of the analytic part of the rule.
1. To find the back pressure in any cylinder, place a suitable diagram factor.
or scale r, against the mean effective pressure in the cylinder, on scale t, and
read result opposite the index arrow line I, on scale t.
Place index arrow line N against the intial pressure on scale y; and, opposite

page_text: the number of expansions on scale w, read result 2 on scale y. Place result* 1,
on scale u against result 2 on scale and read the back pressure, opposite index
arrow line L, on scale .
2. To find the initial pressure in the next cylinder, proceed as in operation 1,
using a different value of diagram factor.
3. To find the number of expansions in the cylinder. Place the suitable
diagram factor, on scale r, against the mean effective pressure, on scale /, and
opposite the index arrow line I read result 1 on scale t. Place index arrow
line L against the back pressure, on scale r, and, opposite result I on scale u
read result 2 on scale r. Place the index arrow line N against the initial pres-
sure, on scale y, and, opposite result 2 on scale y, read the number of expansion3
on scale .
4. To find the initial load on the piston, and reduce it to equivalent load on
the low pressure piston. Determine the back pressure and. initial pressure by
operations 1 and 2; then place the back pressure on scale u, against the initial
pressure on scale ; and, opposite the index arrow line L, read result 1, on
scale. .
This is the effective initial load on the piston.
1
Next place the ratio of this cylinder to the low pressure cylinder, on scale <,
against result 1, on scale ; and read the initial load referred to the low, pressure
piston opposite index arrow line O on scale a.
5. To find the range of temperature in any cylinder. Place the index, arrow
line against the initial pressure 10 on scale y (marked with one arrowhead),
place the back pressure on scales, against the initial pressure, on scale; and,
opposite the index arrow line L, read the range of temperature in degrees
Fahrenheit on scale v.
By means of the synthetic part of my rule the following series of calculations
and inversions of these, involved in the provisional design and adjustment of
cylinder proportions for multiple expansion engines, can be rapidly made without
reference to tables and without the performance of arithmetical work, a valuable
saving of time and labour being effected:-
The determination of the referred mean pressure from the initial pressure,
number of expansions, back pressure, and diagram or efficiency factor.
The determination of the diameter of the low pressure cylinder from the
referred mean pressure, piston speed, and total indicated horse power.
In the case of a triple expansion engine with two low pressure cylinders, the
determination of the diameter of each of the low pressure cylinders from the
single low pressure diameter.
.
The determination of the diameter of the high pressure cylinder from the low
pressure diameter and the cylinder ratio.
The determination of the diameter of intermediate cylinders of triple and
quadruple expansion engines from the cylinder ratios and diameters of the
high pressure and low pressure cylinders.
The determination of the cut off in the high pressure cylinder from the num-
ber of expansions and the total cylinder ratio.
By means of the analytic part of my rule, the following series of analytic
operations connected with the power distribution in multiple expansion engines
can be rapidly accomplished without reference to tables or the performance of
any arithmetical work, thus saving much time and labour:-
The determination of the back pressure in any cylinder from the initial pres-
sure, mean effective pressure number of expansions and diagram factor.
The determination of the initial pressure in the next succeeding cylinder, from
the mean effective pressure, initial pressure, number of expansions and suitable
diagram factor.
The determination of the number of expansions, for adjustment of cut off,
from the initial pressure, back pressure mean effective pressure and suitable
diagram factor.

page_text: The determination of the initial load on the piston of any cylinder from the
initial and back pressures and the further determination of its equivalent load
on the low pressure piston from the cylinder ratio.
The determination of the range of temperature in any cylinder from the
initial and back pressures.
my
Having now particularly described and ascertained the nature of said
invention and in what manner the same is to be performed, I declare that what
I claim is-
1. A calculating rule consisting of a fixed portion and sliders or of two or
more discs or rings or cylinders movable relatively to each other, provided with
the combination of scales and index arrow lines substantially as and for the
purposes herein before described with reference to Figure 1.
2. A calculating rule comprising essentially a combination of scales each
representing a series of values of one of the various factors from which the
total indicated horse power of a multiple expansion engine can be computed
(without direct reference to the particular distribution of this total power in
the various cylinders), the scales being arranged and adapted to be moved
relatively so that different values of the various factors can be chosen at will
and dealt with, and the resulting horse power, for the chosen cylinder pro-
portions, read off; or, conversely, so that the suitable cylinder proportions for
the chosen indicated horse power can be determined, substantially as described
with reference to Figure 1.
3. A calculating rule comprising three pairs of relatively moving parts: the
first pair having on one of its parts a logarithmic scale such as a, a logarithmic
scale such as, an equal division scale such as c and an index arrow line such
as B, and on its other part a logarithmic scale such as d, an index arrow line
such as and an equal division scale such as e; the second pair having on one
of its parts an index arrow line such as G, a logarithmic scale such as f, a
logarithmic scale such as g and an index arrow line such as A, and on its other
part a logarithmic scale such as h, an index arrow line such as 1) and a logarith-
mic scale such as k; and the third pair having on one of its parts an index
arrow line such as F, an index arrow line such as K, a logarithmic scale such as i,
a logarithmic scale such as and a logarithmic scale such as m, and on its other
part a logarithmic scale such as n, a logarithmic scale such as o, a logarithmic
scale such as and an index arrow line such as C; substantially as and for the
purpose described with reference to Figure 1.
4. A calculating rule consisting of a fixed portion and sliders, or of two
or more discs or rings or cylinders movable relatively to each other, provided
with the combination of scales and index arrow lines, substantially as and for
the purposes hereinbefore described with reference to Figure 2.
5.A calculating rule comprising essentially a combination of scales each
representing a series of values of one of the indicated horse-power factors, or
its constituent elements, or the constituent elements of the factors of cylinder
proportions, the scales being arranged and adapted to be relatively moved so
that the more complex factors can be separated into their constituent elements;
and, by taking different values for the factors themselves, the necessary elements
can be determined, and an analysis of the conditions made for any chosen dis-
tribution of the total power in the various cylinders, substantially as described
with reference to Figure 2.
6. A calculating rule comprising three pairs of relatively moving parts; the
first having on one of its parts a logarithmic scale such as q, a logarithmic scale
such as and an index arrow line such as I, and on its other part a logarithmic
scale such ass and a logarithmic scale such as t; the second pair having on one
of its parts an index arrow line such as L and an equal division scale such as u,
and on its other part an equal division scale such as ; and the third pair having
on one of its parts an index arrow line such as M, a logarithmic scale such as w

page_text: an index arrow line such as N and a logarithmic scale such as r, and on its other
part a logarithmic scale such as y, a logarithmic scale such as and an index arrow
line such as ; substantially as and for the purposes described with reference
to Figure, 2.
His 20th day of April 19.
MARKS & CLERK,
18, Southampton Buildings, London, W.C.,
13, Temple Street, Birmingham, and
25, Cross Street, Manchester, Agents.
U.
MARKS & CLERK,
18, Southampton Buildings, London, W.C.,
13, Temple Street, Birmingham, and
25, Cross Street, Manchester, Agents.