An Improved Arithmetical Instrument.
Year: 1894

page_text: Date of Application,
July, 1934
Complete Specification Left, 23rd Apr., 1895-Acc-pted, 15th June, 1895
DALL UT
Complete Specification Left, 23rd Apr., 1895-Acc-pted, 15th June, 1895
PROVISIONAL SPECIFICATION.
An Improved Arithmetical Instrument.
I, GEORGE THOMSON, Plumber 3 Abbey Strand Holyrood Edinburgh do
hereby declare the nature of this invention to be as follows:-
This invention has for its object the arranging of Napier's bones in book form,
in a more improved manner than that described by Sawyer, in Specification 2885.
of the year 1877.
33
In the said Specification the digits of a number are merely placed alongside one
another; after the ordinary manner of writing down numbers: whereas in my
arrangement, the numbers are represented with the "tens figure above the
"units" figure. Fig. 1 (of the accompanying drawings) shews Sawyer's way of
placing the digits; and Fig. 2 shews my way.
One advantage gained by my method of placing the digits, is that instead of
the arithmetical results requiring to be read off in a zig-zag diagonal direction;
the said results are read off in a perpendicular column: and as the figures are all
in one line; it is easier for the eye to follow them, for the purpose of reading the
result.
Another advantage that I gain by my method of placing the digits, as aforesaid;
is, that I am enabled to use short broad strips, instead of long narrow strips. It
will be seen, by reference to Sawyers drawing; that, the strips are about
12 times as long as they are broad. This renders the said strips very liable to be
deranged or torn. This proportion cannot be much reduced in Sawyer's arrange-
ment as the figures require to be made as large, or nearly as large, as the width
of the strips. If however, the digits of the numbers be placed one above the other,
as in my arrangement aforesaid; very small figures may be used in combination
with broad strips: thus rendering the book more manageable, and less liable to
derangement.
Another improvement is effected by having the numbers printed on both sides of
the strips, instead of on one side only; as in Sawyer's arrangement. When
Sawyers book is opened: all the numbers are on the right hand page of the folio ;
while the left hand page is left blank. In my arrangement, the numbers are
distributed over both pages of the folios thus enabling both the length and
breadth of the strips to be reduced; without reducing the size of the figures, or
without reducing the width of the columns. This is of special advantage when it
is desired to make the instrument on a small scale.
I will now describe the construction of my arrangement.
A book (made up of 5 sheets of paper, stitched in the centre to a stiff cover) is
ruled with perpendicular lines to hold 5 columns of figures on each page (The
lines are also ruled on the inside of the covers.) The pages of the book are cut
horizontally into strips: thus forming a number of small books, contained in one
cover. In each of the said small books, figures are arranged as follows:-
The first folio of each of these little books (that is, the folio formed by the
cover, and the first page of the said book) is left blank. In the next folio of eacli
book, a cipher is placed in each of the ten columns, close to the lower edge of. the
leaf. (See Fig. 3.) In the next folio of each book, the columns contain the

page_text: "
numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, in succession; the said figures being placed close
to the lower edge of the leaf. (See Fig. 4.) In the next folio of each book, the
columns contain the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 2, with the " units
figure placed close to the lower edge of the leaf, and with the "tens" figure placed
close to the upper edge; (as shewn in Fig. 5). In all the succeeding folios, the
"tens" and "units" figures are placed in a similar position; that is to say :-
In the next folio of each book, the numbers 3, 6, 9, 12, 15, 18, 21, 24, 27, 3, are
placed as shewn in Fig. 6. In the next folio of each book, the columns contain
the numbers 4, 8, 12, 16, 20, 24, 28, 32, 36, 4, placed as shewn in Fig. 7. In the
next folio of each book, the columns contain the numbers 5, 10, 15, 20, 25, 30, 35,
40, 45, 5, placed as shewn in Fig. 8. In the next folio of each look, the columus
contain the numbers 6, 12, 18, 24, 30, 36, 42, 48, 54, 6, placed as shewn in Fig. 9.
In the next folio of each book, the columns contain the numbers 7, 14, 21, 28, 35,
42, 49, 6, 63, 7, placed as shewn in Fig. 10. In the next folio of each book, the
columns contain the numbers 8, 16, 24, 32, 40, 48, 56, 64, 72, 8, placed as shewn in
Fig. 11.
In the next folio of each book, the columns contain the numbers 9, 18, 27,
36, 45, 54, 63, 72, 81, 9, placed as shewn in Fig. 12. This last folio is the one
formed by the lust leaf of the hook, and the cover which encloses all the books.
A horizontal line may be drawn between the "tens" an "units" figures; but
if there is a wide space between the said figures, the said line will be unnecessary,
and may be dispensed with.
"
The cover is extended downwards a little below the edge of the lowest book, for
the purpose of containing the figures which indicate the numbers of the columns.
The said columns are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, N. The figures used for
numbering the columns are of a different and more conspicous print than the other
figures, for distinction.
The figures in column N indicate the numbers of the folios.
The manner of using the instrument may be understood by the following
example.
It is required to multiply 2457 by 3.
To do this-The first book is opened at folio 2, the next at folio 4, the next at
folio 5, and the next at folio 7. The number 2457 now appears in column N, (as
shewn in Fig. 13). When this is done; one figure of the answer will be found at
each ent line in column 3, (3 being the multiplier) by adding the figure below the
said line, to the figure above the said line.
Thus in column 3 (beginning at the bottom) the first figure we come to is 1; and
as this is the only figure standing at that cut line; the said 1, is the first right
hand figure of the answer. The figures 2 and 5, standing together at the next
cut line (one on each side of it) do, when added together, supply the next figure of
In like manner the figures 1 and 2 standing at the next cut line)
supply the next figure of the answer. And so on.
the answer.
Reading off the answer therefore, in this way, the result is that; on passing the
eye up column 3, and mentaly adding as aforesaid; we read :—
One, seven, three, seven.
Which is written down backwards thus:-
7371.
If two figures, which being added together at a cut line, make more than 9; the
unit only is read and the "tens" figure, (which is always 1), is carried, and added
to the figure, or figures, at the cut line above.
To facilitate turning up the strips, an index may be used. But when this is
done; I prefer to have an index that will not needlessly add to the length of the
strips. For this reason I use an index of the form shown in Fig. 14; or of that
shewn in Fig. 15; in preference to the one which Fig. 16 represents: the latter
being the index described by Sawyer in his Specification.
When however, my improved instrument is constructed on a small scale, the
index may be dispensed with; without the loss of much material advantage;
If two figures, which being added together at a cut line, make more than 9; the
unit only is read: and the "tens" figure, (which is always 1), is carried, and added
to the figure, or figures, at the cut line above.
To facilitate turning up the strips, au index may be used. But when this is
done; prefer to have an index that will not needlessly add to the length of the
strips. For this reason I use an index of the form shown in Fig. 14; or of that
shewn in Fig. 15; in preference to the one which Fig. 16 represents the latter
being the index described by Sawyer in his Specification.
When however, my improved instrument is constructed on a small scale, the
index may be dispensed with; without the loss of much material advantage;

page_text: because (owing to the shortness of the strips) the said strips have a tendency to
spring apart a little; thus enabling the figures, in the N columns, to be seen.
Dated this 26th day of July 1894.
GEORGE THOMSON.
1. GEORGE THOMSON Plumber 3 Abbey Strand Holyrood Edinburgh 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:
This invention has for its object the arranging of "Napiers bones" in book
form, in a more improved manner than that shewn by John Sawyer, in Specification
No. 2885, of the year 1877.
In the said Specification, the digits of a number are merely placed alongside one
another; after the ordinary manner of writing down numbers: whereas in my
arrangement, each number is represented with the " tens figure standing above the
"units" figure.
"
Fig. 1 (of the drawing left with the Provisional Specification) shews Sawyer's
way of placing the digits; and Fig. 2 shews my way.
One advantage gained by my method of placing the digits, is that, instead of
the arithmetical results requiring to be read off in a zig-zag diagonal direction,
(ns in Sawyer); the said results are read off in a perpendicular column: and, as the
figures are all in one line; it is easier for the eye to follow the said figures, for the
purpose of reading the said results.
Another advantage gained by my method of placing the digits as aforesaid, is:
that I am enabled to use short broad strips, instead of long narrow strips. It will
be seen, by reference to Sawyer's drawing, that his strips are about 12 times as long
as they are broad. This renders the said strips very liable to be deranged or torn.
This proportion cannot be much improved, in Sawyer's arrangement; because, the
figures require to be made as large, or nearly as large, as the width of the strips:
(n condition that does not apply to my method of placing the digits). My method
of placing the digits, as aforesaid, admits of the use of short broad strips; because,
if the figures be made small enough to suit the length of the strips; (so that all the
required numbers may be accomodated thereon); it does not matter what interval
(resulting from the breadth of the said strips) there is between the "tens and
"units" figure of each number.
:
73
I therefore desire it to be understood that, the placing of the digits, one above
the other, as shewn in Fig. 2, (for the purpose of rendering the book more
manageable, and less liable to derangement; as well as for the purpose of
simplifying the reading of the arithmetical results) is the essential feature of the
invention.
Another improvement is effected, by having the numbers printed on both sides of
the strips; instead of on one side only, as in Sawyer's arrangement; (page 3,
lines 4 to 11; also page 4, lincs 49 and 50). When Sawyer's book is open; all the
numbers are on the right hand page of the folio; while the left hand page is blank.
In my arrangement, the numbers are distributed over both pages of the lolio. This
enables both the length and breadth of the strips to be reduced; without reducing
the size of the figures; or without reducing the width of the columns. This is of
special advantage, when it is desired to make the instrument on a small scale.
1 will now describe the construction of my arrangement.
A book (made up of 5 sheets of paper stitched in the centre to a stiff cover) is
ruled, at regular intervals with perpendicular lines, to hold 5 columns of figures on each.

page_text: page. (The lines are also ruled on the inside of the covers.) The pages of the
book are cut horizontally into strips; thus forming a number of small books
contained in one cover. In cach, of the said small books, figures are arranged as
follows:
The first folio, of each of these little books, (that is, the folio formed by the
cover and the first page of the said book) is left blank. In the next folio of each
book, a cipher is placed in each of the ten columns, close to the lower edge of the
leaf (see Fig. 3). (All the diagrams, herein refered to, are shewn on the sheet
left with the Provisional Specification.) In the next folio of each book, the
columns contain the numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, in succession; the said
figures being placed close to the lower edge of the leaf: (as shewn in Fig. 4). In
the next folio of each book, the columns contain the numbers, 2, 4, 6, 8, 10, 12,
14, 16, 18, 2; having the "units" figures placed close to the lower edge of the leaf;
and having the " tens figures placed close to the upper edge of the same: (as
shewn in Fig. 5).
"
"
In all the succeeding folios, the "tens and "units" figures are placed in a
similar position; that is to say:-
In the next folio of each book, the numbers, 3, 6, 9, 12, 15, 18, 21, 24, 27, 3, are
placed as shewn in Fig. 6. In the next folio of each book, the columns contain
the numbers, 4, 8, 12, 16, 20, 24, 28, 32, 36, 4, placed as shewn in Fig. 7. In the
next folio of each hook, the columns contain the numbers, 5, 10, 15, 20, 25, 30,
35, 40, 45, 5; placed as shewn in Fig. 8. In the next folio of each book, the columns
contain the numbers, 6, 12, 18, 24, 30, 36, 42, 48, 54, 6; placed as shewn in Fig. 9.
In the next folio of each book, the columns contain the numbers, 7, 14, 21, 28, 35,
42, 49, 56, 63, 7; placed as shewn in Fig. 10. In the next folio of each book, the
columns contain the numbers, 8, 16, 24, 32, 40, 48, 56, 64, 72, 8; placed as shewn in
Fig. 11. In the next folio of each book, the columns contain the numbers,
9, 18, 27, 36, 45, 54, 63, 72, 81, 9; placed as shewn in Fig. 12.
This last folio is the folio formed by the last page of the book, and the cover
which encloses all the books.
This utilising of the cover, as part of the last folio, is an improvement in detail;
because, it reduces the number of seperate moving parts, by dispensing with some
of the strips.
A horizontal line may be drawn between the "tens" and "units" figures; but, if
there is a wide space between the said figures; the said line will be unnecessary,
and may be dispensed with.
The cover is extended downwards below the edge of the lowest book; for the
purpose of containing the figures, which indicate the numbers of the columns. The
said columns are numbered, 1, 2, 3, 4, 5, 6, 7, 8, 9, N: (the letter N standing for
"Number column"). The figures, used for numbering the columns, are of a
different and more conspicious print than the other figures; for distinction.
The figures, in the columns marked N, indicate the numbers of the folios.
The manner of using the instrument may be understood from the following
example
:―
It is required to multiply 2457 by 3.
To do this: The first book is opened at folio 2. the next at folio 4, the next at
folio 5, and the next at folio 7. The number, 2457, now appears in column N: (as
shown in Fig. 13).
When this is done; one figure, of the answer, will be found at each cut line in
column 3, (3 being the multiplier); by adding the figure below the said line, to the
figure above the said line.
:
Thus in column 3 (beginning at the bottom), the first figure we come to is 1;
and as this is the only figure standing at that cut line; the said 1 is the first
right hand figure of the answer. The figures, 2 and 5, standing together at the
next cut line, (one on each side of the said line) do, when added together, supply
the next figure of the answer. In like manner; the figures, 1 and 2, (standing
together at the next cut line) supply the next figure of the answer: and so on.

page_text: Which is written down backwards, thus :-
7371
If two figures, which being added together at a cut line, make more than 9; the
"unit" only (that is the unit, of the number resulting from the addition of the said
two figures) is read; and the "tens" figure (which is always 1) is carried and
added to the figure, or figures, at the cut line above.
For instance; column 7 (see Fig. 13) should be read thus:-
Nine, nine, one, seven, one.
lustead of
Nine, nine, eleven, six, one.
To faciliate turning up the strips, an index may be used. But when this is
done; I prefer to use an index that will not needlessly add to the length of the
strips. For this reason, I use an index of the form shewn in Fig. 14; or of that
shewn in Fig. 15; in preference to the one which Fig. 16 represents the latter
being the index described by Sawyer in his Specification. (See Sawyers
Specification, page 5, lines 1, 2, and 3.)
The index may be constructed as follows:-
A figure (representing the number of the folio on which it stands) is placed in
the index column of each folio. The said index column is a broad space at the right
band side of column N. The reverse side of the paper (on which no index figure is
marked) will be a broad space, on the left hand of the next folio; the said space
being clear of column 1.
The index figures are marked on one side of the paper only; that is, in the
column to the right of column N.
The position of each index figure, on its respective folio, is such; that, were the
strips transparent, the said figures would appear arranged as shewn in Fig. 14, or as
shewn in Fig. 15. The figures of the index being arranged as aforesaid; pieces
are cut off from the various strips as follows:-
A piece is cut out of folio 8; to expose the index figure, 9, (on folio 9). A piece
is cut out of folio 7; to expose the index figures, 8 and 9. A piece is cut out of
folio 6; to expose the index figures, 7, 8, and 9. A piece is cut out of folio 5; to expose
the figures, 6, 7, 8, and 9. A piece is cut out of folio 4; to expose the figures, 5, 6,
7, 8, and 9.
A piece is cut out of folio 3; to expose 4, 5, 6, 7, 8 and 9.
A piece
is cut out of folio 2; to expose 3, 4, 5, 6, 7, 8 and 9. A piece is cut out of folio 1;
to expose 2, 3, 4, 5, 6, 7, 8 and 9. A piece is cut out of folio 0; to expose 1, 2, 3, 4,
5, 6, 7, 8 and 9. A piece is cut off the blank folio; to expose all the index figures.
(See Figs. 14 or 15.)
When, however, my improved instrument is constructed on a small scale: the
index may be dispensed with, without the loss of much material advantage; because,
(owing to the shortness of the strips) the said strips have a tendency to spring
apart a little; thereby allowing the figures, in the "N" columns, to be seen.
Having now particularly described and ascertained the nature of my said inven-
tion, and in what manner the same is to be performed, I declare that what I
claim is:
1. The improved instrument substantially as described.
"
2. In an arrangement of " Napier's bones in book form; placing the digits,
one above the other, as shewn in Fig. 2, for the purposes set forth in the
Specification.
Dated this Twenty-second day of Auril 1895
GEORGE THOMSON.