IMPROVED MEANS FOR ARRIVING AT THE RESULTS OF MULTIPLICATION AND OTHER PROCESSES IN ARITHMETIC.
Year: 1878

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
MULTIPLICATION AND OTHER PROCESSES IN ARITHMETIC.".
Sealed the 12th July 1878, and dated the 24th April 1878.
PROVISIONAL SPECIFICATION left by the said John Sawyer at the Office
of the Commissioners of Patents on the 24th April 1878.
"
JOHN SAWYER, of 3, Adelaide Place, London Bridge. IMPROVED MEANS FOR
ARRIVING AT THE RESULTS OF MULTIPLICATION AND OTHER PROCESSES IN
ARITHMETIC."
The object of the Invention is mainly to arrive at the results of the multiplica-
tion of numbers without calculation, and without the necessity of writing down
figures.
The Invention consists in certain arrangements 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 arrangements of the figures are as follows:-The figures are placed on
separate pieces or strips of paper, linen, cardboard, parchment, wood, metal, or any
other suitable material. The numbers 1, 2, 4, 8, are placed on one piece or strip
and aro repeated constantly, the four figures being carried downwards, and each of
the sets under the first set being preceded by a cipher. The numbers 2, 4, 8,
and 16 are placed on another piece or strip and are repeated constantly, the figures
being carried downwards, each of the single figures in the sets under the first set
being preceded by a cipher. The other pieces or strips are composed of the
following sets of figures, the same mode of arrangement and addition of ciphers
being adopted in each case :-3, 6, 12, and 24; 4, 8, 16, and 32; 5, 10, 20, and 40;

page_text: 6, 12, 24, and 48; 7, 14, 28, and 56; 8, 16, 32, and 64; 9, 18, 36, and 72. On
another piece or strip are placed sets of cyphers, namely, 00, 00, 00, and 00, which
are arranged under each other in the same way; but it is not necessary to have a
set of cyphers at the top of the first row, and that space may therefore be left
blank.
The above-mentioned ten pieces or strips constitute the first row of pieces or
strips, and beyond them to the right are placed as many rows as may be required,
each row being the same as the first row except that each single figure in the top
set also is preceded by a cypher.
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 top figure on the
respective strips, and the strips being so arranged that the index figure on each
strip is exposed to view.
The sets of Figures on the respective strips may be so arranged that the first set
in any particular row shall stand over the second set of the next row to the left,
and the second set in any particular row shall stand over the third set of the next
row to the left, and so on as regards each set downwards; or the sets of figures on
the respective strips may be so arranged that the first set in any particular row
shall stand somewhat to the right of the second set of the row to the left, and the
Becond set in any particular row shall stand the same distance to the right of the
third set of the next row to the left, and so on as regards each set downwards.
If the last-named arrangement is adopted it is advisable to distinguish the
alternate sets of each strip by different colors, the first set being black, the second
red, the third black, the fourth red, and so on; and further, to vary the kind of
figure used, as, for instance, by making the ten's figure of the first set small and the
unit's figure large, the ten's figure of the second set large and the unit's figure small,
the ten's figure of the third set small and the unit's figure large, and so on. By
either of these plans the object is attained of adding up with ease the ten's figures
of the sets in any particular strip, which may be exposed to view with the unit's
figures of the sets in the strip which stands immediately to the left.
The same object may be attained, however, by drawing perpendicular lines
between the ten's and unit's columns of the respective sets, the lines being curved
at the ends to meet the lines in the adjoining upper or under sets.
In that case the
use of cyphers may be dispensed with.
In front of the rows of pieces or strips, herein-before described, are placed in a
horizontal position other strips, either of paper, linen, cardboard, parchment, wood,
metal, or any other suitable material. On the first strip are placed sets of double
cyphers, so arranged as to fall over the first line of figures on the rows (except the
first row) of pieces or strips before described, the cyphers being of the same color
and character of type as the figures in the said rows, and at the right-hand end of
the horizontal strip is placed the index figure 1. The next horizontal strip has the
index figure 2, the third horizontal strip has the index figure 4, and the fourth
horizontal strip has the index figure 8; in other respects the second, third, and
fourth horizontal strips are precisely similar to the first, the cyphers being arranged
to fall over the second, third, and fourth lines of figures in the first-described rows
of pieces or strips.
Beneath the last-described four horizontal strips are placed four others, in all
respects the same except that there are additional double cyphers to fall over the
figures in the second set of figures in the aforesaid rows; but if perpendicular
lines are drawn to correspond with those above described the cyphers may be
dispensed with.
It is advisable to have the index figures of a different color than that of the
first four horizontal strips, other sets of horizontal strips sufficient to cover all the
first-described rows of pieces or strips are continued downwards, each set being
precisely similar to the four strips last described, the index figures being alternately
black and red.

page_text: It will be observed that the firstly-described rows of pieces or strips are composed
of the figures 1, 2, 3, 4, 5, 6, 7, 8, and 9 multiplied by 1, 2, 4, and 8 respectively.
Instead thereof other multiples might be taken in lieu of those of 1, 2, 4, and 8,
as of 1, 2, 3, and 4, the object being to obtain any number up to 9 out of the
figures selected; or the multiples of each of the numerals up to 9 might be taken,
four only having been selected with a view to compactness. The horizontal strips
would of course have to be varied to suit the altered multiples.
The multiplication of numbers is effected in the following manner :-Multiply
392 by 267. Expose to view in the first-described rows of pieces or strips the
following: In the first row the strip having the index 2, in the second row the
strip having the index 6, and in the third row the strip having the index 7. The
whole of the rows of pieces or strips must now be covered by the horizontal strips
with the exception of the following:-The strips indexed 1 and 2 of the first set of
four, the strips indexed 1 and 8 of the second set of four, and the strip indexed 2
of the third set of 4. The figures now exposed to view are,

page_text: SPECIFICATION in pursuance of the conditions of the Letters Patent filed by
the said John Sawyer in the Great Seal Patent Office on the 24th October
1878.
JOHN SAWYER, of 3, Adelaide Place, London Bridge. "IMPROVED MEANS FOR
ARRIVING AT THE RESULTS OF MULTIPLICATION AND OTHER PROCESSES IN
ARITHMETIC."
The object of the Invention is mainly to arrive at the results of the multiplica-
tion of numbers without calculation, and without the necessity of writing down
any figures except those representing the result.
The Invention consists in certain arrangements of strips and of figures marked
upon them, as herein-after described.
The numbers 1, 2, 4, 8, are placed on one strip one below the other, commencing
at the upper end of the strip, and they are repeated for any desired number of
times, the four figures being carried downwards, as shewn at Figure 1 of the
Drawings hereunto annexed, and at the bottom of this strip is placed the index
figure 1.
On another strip are placed the multiples of these numbers with 2,
viz., 2, 4, 8, 16, and these figures are similarly placed one below the other on the
strip, and repeated, as shewn at Figure 2, and at the bottom of this strip is placed
the index figure 2. Other pieces or strips similarly have repeated upon them the
multiples of the numbers 1, 2, 4, 8, with 3, with 4, &c., up to 9, and across each
strip are drawn diagonal lines to divide the units from the tens, as shown at
Figure 2. The strips are placed one above the other, the strip having upon it the
index figure 1 being uppermost. Above this is also another strip with sets of
cyphers upon it, as shewn at Figure 3, and above this there may be a plain or
colored strip with no figures upon it; Figure 3 also shews that the several strips
are successively one longer than the other, so that the index figures at the lower
end of each may all be seen.
The above-mentioned ten strips constitute the first set or pile of strips, and
beyond them to the right are placed as many similar sets or piles as may be required.
The several sets or piles thus placed side by side will appear as shewn at Figure 4.
Eight such rows are shewn in this figure, but the number may be varied according
as it is desired to provide for multiplying a greater or less number of figures.
It is desirable that the rows of figures between the several diagonal lines across
the strips should be alternately in different colors. The top row on the first set
of strips, say red, the second black, the third red, and so on. The top row on the
second set of strips would then commence with black, and the next row would
be red, and so on. This is indicated at Figure 4, dotted lines being used to
denote red.
By the above arrangements the ten's figures, on any particular strip which may
be exposed to view, may easily and without chance of error he added to the exposed 4
units figures on the strip which stands immediately to the left of it.
Other means than those mentioned might be adopted for distinguishing the
units from the tens on each strip, but those described answer the purpose well.
Above the several sets of strips arranged as herein-before described and shewn
at Figure 4, are placed other horizontal strips, as shewn at Figure 5. These strips 4
are of a width each to cover over one horizontal line or row of figures from the
several vertical strips.
The
At the right-hand end of the first horizontal strip is the index figure ], on the
next the index figure 2, the next 4, the next 8, the next again 1, and so on.
other ends of the strips are held in any suitable manner, so that they can be turned 5
back to expose any part of the vertical strips, the upper ends of the vertical strips
are similarly held in such a way that they can be turned back, so that any one or
other of the strips in each pile or set may be exposed to view.
I will now give an example of the manner in which the apparatus above
described is used for multiplication.
5

page_text: To commence with, the horizontal strips are all turned back out of the way.
The slip 7 of the first, that is the left-hand set of vertical strips, is exposed to view
by turning back the strips above it. The strip 4 of the next vertical set is
similarly exposed to view, the strip 8 of the next set, and so on for all the
figures 7, 4, 8, 6, 2, 9, 3. The horizontal strips are then all turned back to cover
over the vertical strips which have thus been exposed, and such of them as are
required to be turned back to expose the figures on the vertical strips below them
are then selected and turned back out of the way. Thus, for the number 846,253,
the first horizontal slip with the index figure 8 upon it is turned back, the next
slip below it with the index figure 4, the next slips 2 and 4 for the figure 6, then
the next with the index figure 2, and so on, as represented at Figure 5. The
figures in each of the several columns between the diagonal lines are then added
together as in ordinary addition. The operation is commenced at the right, thus,
6+3=9, this is the unit figure of the product, then 8 +9 +2+3
22, put
down 2 for the second figure and carry 2 to the next column, and so on for all the
several columns of figures until the answer 6,335,297,910,129 is obtained.
In the apparatus above described, the several rows of figures on the strips in
each set or pile are composed of the figures 1, 2, 3, 4, 5, 6, 7, 8, and 9 multiplied.
by 1, 2, 4, and 8 respectively. Instead thereof other multiples might be taken, as,
for example, 1, 2, 3, and 4, the object being to enable any multiple up to 9 to be
made up from the multiples selected; thus, if the multiples 1, 2, 3, and 4 be used,
5 may be obtained by exposing the 3 and 2, 6 by exposing the 4 and 2, and so on;
or the multiples of each of the numerals up to 9 might be taken. I however prefer
to use four only, so as to make the apparatus more compact.
In place of making the several strips to turn or fold back at one of their ends,
as above described, various mechanical or other devices might be adopted for
exposing or covering the number in the manner required. The strips also may be
of paper, linen, cardboard, parchment, wood, metal, or other suitable material.
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 the multiplication of numbers without
the necessity of writing down any figures except those representing the result.
In witness whereof, I, the said John Sawyer, have hereunto set my hand
and seal, this Sixteenth day of October, in the year of our Lord One
thousand eight hundred and seventy-eight.
JOHN SAVED