IMPROVED MEANS FOR ARRIVING AT THE RESULTS OF MULTI- PLICATION AND OTHER PROCESSES IN ARITHMETIC
Year: 1877

page_text: LETTERS PATENT to John Sawyer, of 3, Adelaide Place, London Bridge, for
the Invention of "IMPROVED MEANS FOR ARRIVING AT THE RESULTS OF MULTI-
PLICATION AND OTHER PROCESSES IN ARITHMETIC."
Sealed the 4th January 1878, and dated the 28th July 1877.
PROVISIONAL SPECIFICATION left by the said John Sawyer at the Office
of the Commissioners of Patents on the 28th July 1877.
JOHN SAWYER, of 3, Adelaide Place, London Bridge. "IMPROVED MEANS FOR
ARRIVING AT THE RESULTS OF MULTIPLICATION AND OTHER PROCESSES IN ARITH-
METIC."
The object of the Invention is to arrive at the results of the multiplication and
division of numbers without the mental labour attendant on the ordinary processes
of multiplication and division.
The Invention consists in a certain arrangement of figures, and in the method
adopted for bringing such figures into any required combination.
The figures may be printed, written, or engraved either on paper, linen, card-
board, parchment, wood, metal, or any other suitable material.
The arrangement of the figures is as follows:-The figures are placed at regular
distances apart from one another on separate pieces or strips of paper, linen, card-
board, parchment, wood, metal, or any other suitable material. The numbers from
1 to 9 inclusive are placed on one piece or strip; the numbers 2, 4, 6, 8, 10, 12, 14,
16, and 18, on another piece or strip; the numbers 3, 6, 9, 12, 15, 18, 21, 24, and 27,
on another piece or strip; the numbers 4, 8, 12, 16, 20, 24, 28, 32, and 36, on
another piece or strip; the numbers 5, 10, 15, 20, 25, 30, 35, 40, and 45, on another
piece or strip; the numbers 6, 12, 18, 24, 30, 36, 42, 48, and 54, on another piece
or strip; the numbers 7, 14, 21, 28, 35, 42, 49, 56, and 63, on another piece or
strip; the numbers 8, 16, 24, 32, 40, 48, 56, 64, and 72, on another piece or strip;
and the numbers 9, 18, 27, 36, 45, 54, 63, 72, and 81, on another piece or strip.
The above-mentioned nine pieces or strips constitute the first row of pieces or

page_text: strips, and beneath them is placed another row of nine strips, with the same
numbers on the respective strips (the numbers which consist of one figure only
being preceded by a cypher).
The cypher may also be placed in the first or top row, and in addition a tenth
piece or strip, with the cypher 0 repeated, and placed in the same positions as the
numbers on the other strips. As many rows of pieces or strips as may be con-
sidered necessary are placed beneath, that is, lower down than the second row, each
successive row containing ten strips, precisely similar to those in the second row.
It is advisable that over the respective numbers in each piece or strip in the
first row should appear in consecutive order the respective numbers from 1 to 9
inclusive.
For facility of reference to any required strip, it is advisable that an index figure
be placed on each strip, the index figure being the same as the first and lowest
number on the respective strips, and the strips being so arranged that the index
figure on each strip is exposed to view.
The rows of strips may be so placed that each row shall have the unit figure of
each number the space of one figure to the right of the unit figure of the number
in the row immediately above, or the figures in each row could be placed perpen-
dicularly beneath each other. In the latter case it would be advisable to show the
connection between the figures by a line drawn from the figure in the units' place
of each number towards the figure in the tens' place of the number in the row
immediately beneath.
Various mechanical means may be adopted for exposing to view the numbers on
any desired strip of any desired row or series, and the addition of the respective
columns of numbers falling beneath each other (on the adjustment in any desired
way) of any two or more of the rows of slips will give the product resulting from
the multiplication by the numbers from 1 to 9 of the number formed by the figures
(excluding such cyphers as stand before numerals) which stand at the extreme left
of the said rows of slips.
SPECIFICATION in pursuance of the conditions of the Letters Patent filed
by the said John Sawyer in the Great Seal Patent Office on the 26th January
1878.
JOHN SAWYER, of 3, Adelaide Place, London Bridge. "IMPROVED MEANS FOR
ARRIVING AT THE RESULTS OF MULTIPLICATION AND OTHER PROCESSES IN ARITH-
METIC."
The object of the Invention is to arrive at the results of the multiplication and
division of numbers without the mental labour attendant on the ordinary processes
of multiplication and division.
The Invention consists in a certain arrangement of figures, and in the method
adopted for bringing such figures into any required combinations.
The figures may be printed, written, or engraved either on paper, linen, card-
board, parchment, wood, metal, or any other suitable material.
"

page_text: The arrangement of the figures is as follows:-The figures are placed at regular
distances apart from one another on separate pieces or strips of paper, linen, card-
board, parchment, wood, inetal, or any other suitable materials. The numbers from
1 to 9 inclusive are placed on one piece or strip; the numbers 2, 4, 6, 8, 10, 12, 14,
16, and 18, on another piece or strip; the numbers 3, 6, 9, 12, 15, 18, 21, 24, and 27,
on another piece or strip; the numbers 4, 8, 12, 16, 20, 24, 28, 32, and 36, on
another piece or strip; the numbers 5, 10, 15, 20, 25, 30, 35, 40, and 45, on another
piece or strip; the numbers 6, 12, 18, 24, 30, 36, 42, 48, and 54, on another piece
or strip; the numbers 7, 14, 21, 28, 35, 42, 49, 56, and 63, on another piece or
strip; the numbers 8, 16, 24, 32, 40, 48, 56, 64, and 72, on another piece or strip;
and the numbers 9, 18, 27, 36, 45, 54, 63, 72, and 81, on another piece or strip.
The above-mentioned nine pieces or strips constitute the first row of pieces or
strips, and beneath them is placed another row of nine strips, with the same
numbers on the respective strips (the numbers which consist of one figure only
being preceded by a cypher).
The cypher may also be placed in the first or top row, and in addition a tenth
piece or strip, with the cypher 0 repeated and placed in the same positions as the
numbers on the other strips. As many rows of pieces or strips as may be con-
each
sidered necessary are placed beneath, that is, lower down than the second row,
successive row containing ten strips precisely similar to those in the second row.
It is advisable that over the respective numbers in each piece or strip in the
first row should appear in consecutive order the respective numbers from 1 to 9
inclusive.
For facility of reference to any required strip, it is advisable that an index figure
be placed on each strip, the index figure being the same as the first and lowest
number on the respective strips, and the strips being so arranged that the index
figure on each strip is exposed to view.
The rows of strips may be so placed that each row shall have the unit figure of
each number the space of one figure to the right of the unit figure of the number
in the row immediately above; or the figures in each row could be placed perpen-
dicularly beneath each other. In the latter case it would be advisable to show the
connection between the figures by a line drawn from the figure in the units' place
of each number towards the figure in the tens' place of the number in the row
immediately beneath.
Various mechanical means may be adopted for exposing to view the numbers on
any desired strip of any desired row or series, and the addition of the respective
columns of numbers falling beneath each other (on the adjustment in any desired
way) of any two or more of the rows of strips will give the product resulting from
the multiplication by the numbers from 1 to 9 of the number formed by the figures
(excluding such cyphers as stand before numerals) which stand at the extreme left of
the said rows of strips.
The Drawing hereunto annexed shews one way in which the rows or sets of strips
may be mounted to enable any one or other of the strips of any of the rows to be
exposed to view. In this arrangement the slips composing each series or row of
Each band is
slips are connected together, so as to form a jointed endless band a.
mounted on a square roller b; each roller b can be turned by a handle at its end, so
that any one or other of the strips composing the endless chain which the roller
carries can be brought uppermost. Light springs may be used to bear against one
or other of the flat sides of the rollers to hold them in any position in which they
may be turned.
The following is the manner in which the apparatus is to be used for multiplica-
tion:-
Example: Multiply 769250361 by 98765432.
The roller carrying the top row or series of strips is turned until the strip with the
figure 7 at its end is brought uppermost; the roller carrying the second series of
strips is turned until the strip with the figure 6 is upperinost, and so on for the
several rollers. The Drawing shows the apparatus with the requisite strips thus

page_text: brought uppermost. The numbers upon the strips under the heading 2 are now
added up as they stand one under the other; the total will be seen almost at a
glance, and is written down. The numbers under the heading 3 are then similarly
added together and written down below the first line, as in ordinary multiplication,
and, as shown below, by adding together the eight lines of figures so written down
the product is obtained.
For division the tables are used in the same way as for multiplication. The
divisor takes the place of the multiplicand, and, like it, having been once found in
the tables is dismissed from the mind. In other respects the process is similar to
ordinary division. The dividend is set down, and dealt with in the usual way.
The divisor must be looked out in the tables, and need not be written down.
Example: Divide 75975344220320952 by 769250361.
The Drawing shows the apparatus set for this divisor.
It is not necessary to consider at each step of the process how many times the divisor
is contained in the figures to be dealt with, as an inspection of the first two rows is
sufficient to show approximately the first three figures. In this example the first
three figures are 759, and a glance at the top rows shows that the "9" column will
give for the first three figures over 685; the total of that column is therefore set
down and deducted in the ordinary way. The first three figures of the remainder
are 674; the top rows of the "8" column show 615, therefore the total of that
column is taken, and so on.
Various other ways than the one above described might be adopted for mounting
the slips in rows or sets, so that any one or other slip of each row or set may be
exposed to view whenever required. For example, a very convenient arrangement
is to bind them together at their left-hand ends, and so keep them together in the

page_text: form of a book. In this case the several slips of each row or set come one below
the other, and are made successively longer one than the other, the lowest being
longest, so that the index figure at the end of each of the slips may be exposed to
view. With this arrangement any one or other of the slips of any of the rows or
sets can be exposed to view by turning back the several slips which are above it, in
a similar way to turning over the leaves of a book.
Having thus described the nature of my Invention, and the manner of performing
the same, I would have it understood that I claim the improved means herein-
before described for arriving at the results of multiplication and other processes in
arithmetic.
In witness whereof, I, the said John Sawyer, have hereunto set my hand
and seal, this Twenty-third day of January, in the year of our Lord One
thousand eight hundred and seventy-eight.
JOHN SAWYER. (LS.)