IMPROVEMENTS IN SLIDE RULES
Year: 1878

page_text: LETTERS PATENT to Frederick Augustus Sheppard, of the Town and County of
Southampton, Civil Engineer, for the Invention of "IMPROVEMENTS IN SLIDE
RULES."
Sealed the 18th October 1878, and dated the 2nd August 1878.
PROVISIONAL SPECIFICATION left by the said Frederick Augustus Sheppard
at the Office of the Commissioners of Patents on the 2nd August 1878.
FREDERICK AUGUSTUS SHEPPARD, of the Town and County of Southampton, Civil
Engineer. "IMPROVEMENTS IN SLIDE RULES."
The object of the above-named improved slide rule, hereafter described as "the
rule," is to facilitate instrumental calculations, bringing the same into easy and
efficient use by artificers, and thereby rendering unnecessary a multiplicity of gauge
points, special divisors, mean areas, and other devices by means of which the form
of rule at present in use is rendered of limited practical value, because, as now con-
structed, its use requires a special study, and its manipulation is tedious.
The foregoing advantages are ensured in rules constructed according to my
Invention, and the body of the rule is made with two fixed or standing parts;
between which, either cut out of the same piece or fastened by glue, cement, or
screws, or any other convenient method, is a concealed guide forming two grooves,
one opposite to the groove in each of the standing parts, the upper surface of the
concealed guide being kept below the plane of the upper surface of the standing
parts. Into the grooves formed by the concealed guide and standing parts work
two slides of such a form as to be reversible, that is to say, the grooves before
mentioned are so contrived as to allow of either surface of the slides being brought
into the same plane as the surface of the standing parts, whether the slides be
reversed or not.
The rule is made of any material that is applicable to the same, and the alides
and grooves are made of such a form as will suit the material of which the rule is
constructed.

page_text: ороско
and concealed guides that the object of improved instrumental calculation can be
obtained thereby.
An upper line on the face is called lengths, superficial area.
A first slide is called breadths.
A second slide is called thickness.
A lower line is called cubical contents.
The rule having been constructed as described, the marking will now be
elucidated.
As it is intended to apply the principle to two systems of calculation, I shall
first refer to the decimal system.
For decimal calculation the markings or divisions on all the parts of the rule
will be the usual logarithmic lines on the surface edges of the standing parts, also
upon the edges of the slides that work in close proximity to the standing parts and
to each other, these markings thus being in duplicate upon each slide, that is to say,
on each edge of the slides.
On the reverse surface of the slides can be drawn the usual logarithmic lines of
square roots upon one and cubic roots upon the other, by means of which the higher
powers of numbers can be obtained.
To shew the facility that the rule gives in working ordinary calculations, take,
for example, the product of 18x7x3=378. Under 18, the length, on the line of
lengths, place 1 on the line of breadths; under 7, the breadth, place 1 on the line of
thickness; then under 3, the thickness, on the line of thickness, is found 378, the
cubic contents.
18 x 7 x 3
27
Supposing the previous example to represent feet, and the result is desired in
cubic yards, the question will stand thus,
14. In this instance, place
under 18, the length on the line of lengths, 27, the divisor, on the line of breadths;
then under 7, the breadth, on the line of breadths, place 1 of the line of thickness;
under 3, of the line of thickness, will be found 14, the cubic yards required.
By these two simple examples it will be seen how the combination here intro-
duced in constructing and marking the rule brings instrumental calculation within
the limit of the ordinary workman's capacity, by dividing the manipulation of all
reference to the gauge point ✓ 27, or the 7×3, as is necessary in working the
common slide rule.
The second system of duodecimals requires the rule to be marked differently,
that is to say, duodecimally, so as to bring the capabilities of the rule within the
power of working in feet and inches, wherein lies a great source of usefulness, for
by the ordinary slide rule feet and inches have to be converted into feet and deci-
mals to work the question, and when the result is obtained the converse operation
has to be performed. The result is that in cubing timber except by the rule of
the length of rough timber multiplied by the square of the quarter girth, the slide
rule is seldom used.
The duodecimal rule is constructed precisely in the same manner as the decimal
rule already described, but the markings are different. The logarithmic divisions
for the larger divisions are retained as commonly used, but the smaller divisions are
marked into 12ths instead of 10ths, yet their divisions retain their logarithmic
character.
The whole of parts of the duodecimal rule are marked on the edges of the standing
parts, and on each edge of the slides, breadth and thickness. By the means thus
adopted the rule can be used for obtaining products in feet and inches, such as are
generally required in the operations of the builder and other artificers. Thus, sup-
posing it be required to find the cubic contents of a log of timber 21' 6". long, 1'.1"
broad, and 1', 1" deep. The manipulation is exactly similar to the use of the
decimal slide rule. Under 21.6", the length, in the line of lengths, place the 1 of
the line of breadths; then under 1.14", the breadth, on the line of breadths, place
the 1 of the line of thickness; then under 1'.1", the thickness, on the line of thick-

page_text: ness, will be found the cubic contents, 26'. 21" Thus the result is obtained directly
in terms of cubic feet and inches, that is to say, the cubic inch is one-twelfth part of
a cubic foot, as is usually done in trade calculations.
Thus it is established that instrumental calculation is facilitated by the combina-
tion of lines and slides wherein the arrangements of the lines, the form of the slides,
and the introduction of the concealed guide is such as has not hitherto been
attempted, and it will be found that an immense number of gauge points or
special divisors are not required, that the results may be obtained from a knowledge
of rules in general use in the arts and sciences, also that no previous working is
required to a great majority of cases, and where once the rule is set to work any
example, then the example remains upon the rule without the necessity of any
shifting, which is a great advantage to operators, rendering it possible to test the
accuracy of the work at all stages of the operation.
SPECIFICATION in pursuance of the conditions of the Letters Patent filed by
the said Frederick Augustus Sheppard in the Great Seal Patent Office on the
1st February 1879.
FREDERICK AUGUSTUS SHEPPARD, of the Town and County of Southampton,
Civil Engineer. "IMPROVEMENTS IN SLIDE RULES."
"
The object of the above-named slide rule, hereafter described as the rule," is
to facilitate instrumental calculations, bringing the same into easy and efficient use
by artificers, and thereby rendering unnecessary a multiplicity of guage points,
special divisors, means, areas, and other devices by means of which the form of rule
at present in use is rendered of limited practical value, because as now constructed
its use requires a special study, and its manipulation is tedious.
The foregoing advantages are ensured in rules constructed according to my
Invention.
Figure 1 of the accompanying Drawings shows a slide rule made according to
this Invention, divided according to the decimal system, and showing the front face
of the slides; Figure 1ª is a similar view, but shewing the reversed faces of the
slides.
Figure 2 is a cross section of the rule.
The body of the rule is made with two fixed or standing parts a, a, between
which, either cut out of the same piece or fastened by glue, cement, or screws, or
any other convenient method, is a concealed guide b, forming two grooves, one
opposite to a groove in each of the standing parts a, a; the upper surface of the
concealed guide b being kept below the plane of the upper surface of the standing
parts a. Into the grooves formed by the concealed guide b and standing parts a, a,
work two slides c and d of such a form as to be reversible, that is to say, the grooves
before mentioned are so contrived as to allow of either surface of the slides'c, d,
being brought into the same plane as the surface of the standing parts a, a, whether
the slides be reversed or not.

page_text: The rule is made of any material that is applicable to the same, and the slides
and grooves are made of such a form as will suit the material in which the rule is
constructed.
The special object of the Invention is to form a rule with such slides and grooves
and concealed guides that the object of improved instrumental calculation can be
obtained thereby, as shewn at Figures 3, 3ª, and 3b, which are sections of the
improved slide rule, Figure 3 being suitable for paper, Figure 3a for wood or iron,
and Figure 3b for metal slides.
On the decimal rule, Figures 1 and 1ª, the upper line on the face is called lengths,
superficial area, and squares.
The first slide c is called breadths on the face shewn in Figure 1, and square roots
and numbers on the reverse face shewn in Figure 1ª.
The second slide'd is called thickness on the face shewn at Figure 1, and numbers
on the reverse face shewn in Figure 1ª.
The lower line is called cubical contents.
Figure 6 shews a rule divided according to the duodecimal system, and shewing
the front face of the slides; Figure 6 is a similar view, shewing the reversed face
of the slide. The section is as Figure 2.
On this duodecimal rule,-
The upper line on the face is called lengths, superficial area, and square roots.
The first slide c¹ is called breadths on the face, and squares and numbers on the
reverse.
The second slide d' is called thickness on the face, and numbers on the reverse.
The lower line is called cubical contents.
The rule having been constructed as described, the markings will now be
elucidated.
As it is intended to apply the principle to two systems of calculations, I shall
first refer to the decimal system, and to Figures 1 and 1ª.
For decimal calculations the markings or divisions on all the parts of the rule will
be the usual logarithmic lines on the surface edges of the standing parts a, a, also
upon the edges of the slides c, d, that work in close proximity to the standing parts
and to each other, these markings thus being in duplicate upon each slide c and d,
that is to say, on each edge of the face of the slides. These markings of the loga-
rithmic scales or radii, as they are usually called, are shewn in duplicate upon each
edge of the standing parts a and slides.c, d, for the purpose of description and
explaining the uses of the rule, but they may be carried to any number of two or
more radii in length, as may be found necessary to secure sufficient accuracy in the
calculations that the rule may be intended to compute.
On the reverse surface of the slides c and d can be drawn the usual logarithmic
lines of square roots upon one and cube roots upon the other by means of which the
higher powers and roots of numbers can be obtained.
To sliew the facility that the rule gives in working ordinary calculations, take,
for example, the product of 18×7x3=378 is to be ascertained. This operation
is illustrated by the diagram, Figure 4, and is as follows:-Under 18, the length,
on the line of lengths, place 1 on the line of breadths; under 7, the breadth, on the
line of breadths, place 1 on the line of thickness; then under 3, the thickness, on
the line of thickness, is found 378, the cubic content, as shewn.
Supposing the previous example to represent feet, and the result is desired in
cubic yards, the question will stand thus, =14. This operation is repre-
sented in the diagram, Figure 5. In this instance, place under 18 feet, the length,
on the line of lengths, 27 (the devisor) on the line of breadths; then under 7,
the breadth, on the line of breadths, place 1 of the line of thickness; under 3
on the line of thickness will be found 14, the cubic yards required, as shewn.
18 x 7 x 8
27
By these two simple examples it will be seen how the combination here intro-
duced in constructing and marking the rule brings instrumental calculation within
the limit of the ordinary workman's capacity, by divesting the manipulation of all

page_text: CACHECO.che guage pot v 27, of the x 0,83 15 necessary in WORKING
common slide rule.
I will now proceed to shew how the reverse sides of the slides of the rule as
shewn in Figure 1 can be adapted to working calculations in addition or subtrac-
tion, or involving the square or square root of a number, or any other power
or root.
The usual line of square roots, commonly called the D line, is placed upon the
upper edge of the upper slide. The lower edge of the upper slide c, and both edges
of the lower slide d, are marked in equal divisions, that is to say, the length of each
radius is divided decimally. By this arrangement that useful and most frequent
question of two sides of a right-angled triangle being given to find the third side;
can be solved by only one movement of the lower slide; the result being the square
root of either the sum or difference of the squares of the sides. Suppose the base
and perpendicular are given equal to 21 and 28, and it be required to find the
bypothenuse. This operation is illustrated by the diagram, Figure 5º. Place the
line of square roots symetrically under the line of lengths 1 to 1: 10 to 10. The
two lines now represent the squares and square roots of numbers. Look for the
square of 28 on the line of squares, over 28 on the square root line, which is found
to be 784, then move the lower slide d until its 0 on the line of numbers be under
784 on the line of numbers on the upper slide. Then look for the square of 21 on
the square line, over 21 on the square root line, which is found to be 441. Then
over 441 on the line of numbers on the lower slide will be found 1225, which is the
sum of the squares of the two sides of the triangle. Then under 1225 on the square
line is found its square root 35, which is the length of the hypothenuse.
Had the side been required from the hypothenuse, and the other side being given,
then the square of the given side would have been deducted from the square of
hypothenuse. Thus, let the hypothenuse 35 and the side 28 be given to find the
other side. This operation is illustrated by the diagram, Figure 5b. Find the
square of 35 as before =1225, and the square of 28 =784, by mere inspection;
then place 784 on the lower slide d under 1225 on the upper slide c; then the o
of the lower slide d will be under 441 on the upper slide c; whose square root is
found by inspection as before to be 21, the length of side required.
The second system of duodecimals is as aforesaid illustrated by Figures 6 and 60;
it requires the rule to be marked differently, that is to say, duodecimally, so as
to bring the capabilities of the rule within the power of working in feet and
inches, wherein lies a great source of usefulness, for by the ordinary slide rule
feet and inches have to be converted into feet and decimals to work the question,
and when the result is obtained the converse operation has to be performed. The
result is that in cubing timber except by the rule of the length of rough timber
multiplied by the square of the quarter girth the slide rule seldom used.
The duodecimal rule is constructed precisely in the same manner as the decimal
rule already described, but the markings are different. The logarithmic divisions
for the larger divisions are retained as commonly used, but the smaller divisions are
marked into 12ths instead of 10ths, yet these divisions retain their logarithmic
character, as shewn at Figure 6 and Figure 6º. Logarithmic feet are divided into
inches logarithmically.
The whole of the parts of the duodecimal rule are marked in the manner described
above, that is to say, on the edges of the standing parts a and b, and on each edge
of the face of the slides c¹ and d', breadth and thickness. By the means thus
adapted the rule can be used for obtaining products in feet and inches, such as are
generally required in the operations of the builder and other artificers. Thus, sup-
posing it be required to find the cubic contents of a log of timber 21' 6" long, 1'. 1"
broad, and 1'. 1" deep. This manipulation is illustrated by the diagram, Figure 7,
and is exactly similar to the use of the decimal slide rule. Under 21'6", the length,
on the line of lengths, place 1 of the line of breadths; then under 1' 11", the breadth,
on the line of breadth, place the 1 of the line of thickness; then under 1'. 1", the

page_text: thickness, on the line of thickness, will be found the cubic content, 26'. 21". Thus
the result is obtained directly in terms of cubic feet and inches, that is to say,
the cubic inch as one twelfth part of a cubic foot, as is usually done in trade
calculations.
The reverse sides of the slides c¹, d³, can be used for a similar purpose as described
for the decimal rule; thus the reverse side of the upper slide c¹ is marked on the
upper edge with the squares of the numbers on the line of lengths, and on the same
principle as the line of lengths, that is to say, the subdivisions are marked loga-
rithmically into 12ths, in accordance with the system of duodecimals. The lower
edge of the reverse of the upper slide c¹ and the two edges of the reverse of the
lower slide d¹ are divided into equal parts, to represent numbers, as described as
appertaining to the decimal rule.
As the working of the reverse of the slides in the duodecimal rule (bearing in
mind the designation of the lines, squares, and square roots) is exactly similar to
the same operation performed on the decimal rule, it is needless to recapitulate
the description.
In the instance of the duodecimal rule the squares and square roots are marked
in 12th instead of 10ths as in the decimal rule.
Thus it is established that instrumental calculation is facilitated by the combina-
tion of lines and slides wherein the arrangements of the lines, the form of the slides,
and the introduction of the concealed guide is such as has not hitherto been
attempted, and it will be found that an immense number of gauge points or
special divisions are not required, that the results may be obtained from a know-
ledge of rules in general use in the arts and sciences, also that no previous working
is required to a great majority of cases, and when once the rule is set to work any
example, then the example remains upon the rule without the necessity of any
shifting, which is a great advantage to operators, rendering it possible to test the
accuracy of the work at all stages of the operation.
And having thus described and ascertained the nature of this Invention, and how
the same is or may be performed, I declare that I claim,-
First. The mode of numerically marking a slide rule, substantially as and for the
purposes set forth.
Second. The construction of a slide rule with one or two or more grooved car-
riers containing two or more slides, reversible or not, and all marked numerically,
as set forth.
Third. The construction of a slide rule with two or more reversible slides, con-
structed and numerically marked as illustrated in the Drawings.
In witness whereof, I, the said Frederick Augustus Sheppard, have hereunto
set my hand and seal, this Twenty-ninth day of January, in the year of
our Lord One thousand eight hundred and seventy-nine.