Those who can perform in that most necessary of all mathematical
operations, simple addition, any great number of successive examples
or any single extensive example without consciousness of a severe
mental strain, followed by corresponding mental fatigue, are
exceptions to a general rule. These troubles are due to the quantity
and complexity of the matter with which the mind has to be occupied at
the same time that the figures are recognized. The sums of pairs of
numbers from zero up to nine form fifty-five distinct propositions
that must be borne in memory, and the "carrying" is a further
complication. The strain and consequent weariness are not only felt,
but seen, in the mistakes in addition that they cause. They are, in
great part, the tax exacted of us by our decimal system of arithmetic.
Were only quantities of the same value, in any one column, to be
added, our memory would be burdened with nothing more than the
succession of numbers in simple counting, or that of multiples of two,
three, or four, if the counting is by groups.
It is easy to prove that the most economical way of reducing addition
to counting similar quantities is by the binary arithmetic of
Leibnitz, which appears in an altered dress, with most of the zero
signs suppressed, in the example below. Opposite each number in the
usual figures is here set the same according to a scheme in which the
signs of powers of two repeat themselves in periods of four; a very
small circle, like a degree mark, being used to express any fourth
power in the series; a long loop, like a narrow 0, any square not a
fourth power; a curve upward and to the right, like a phonographic
_l_, any double fourth power; and a curve to the right and downward,
like a phonographic _r_, any half of a fourth power; with a vertical
bar to denote the absence of three successive powers not fourth
powers. Thus the equivalent for one million, shown in the example
slightly below the middle, is 2^{16} (represented by a degree-mark in
the fifth row of these marks, counting from the right) plus 2^{17} +
2^{9} (two _l_-curves in the fifth and third places of _l_-curves)
plus 2^{18} + 2^{14} + 2^{6} (three loops) plus 2^{19} (the _r_-curve
at the extreme left); while the absence of 2^{3}, 2^{2}, and 2^{1} is
shown by the vertical stroke at the right. This equivalent expression
may be verified, if desired, either by adding the designated powers of
two from 524,288 down to 64, or by successive multiplications by two,
adding one when necessary. The form of characters here exhibited was
thought to be the best of nearly three hundred that were devised and
considered and in about sixty cases tested for economic value by
actual additions.
In order to add them, the object for which these forty numbers are
here presented in two notations, it is not necessary to know just
_why_ the figures on the right are equal to those on the left, or to
know anything more than the order in which the different forms are to
be taken, and the fact that any one has twice the value of one in the
column next succeeding it on the right. The addition may be made from
the printed page, first covering over the answer with a paper held
fast by a weight, to have a place for the figures of the new answer as
successively obtained. The fingers will be found a great assistance,
especially if one of each hand be used, to point off similar marks in
twos, or threes, or fours--as many together as can be certainly
comprehended in a glance of the eye. Counting by fours, if it can be
done safely, is preferable because most rapid. The eye can catch the
marks for even powers more easily in going up and those for odd powers
(the _l_ and _r_ curves) in going down the columns. Beginning at the
lower right hand corner, we count the right hand column of small
circles, or degree marks, upward; they are twenty-three in number.
Half of twenty-three is eleven and one over; one of these marks has
therefore to be entered as part of the answer, and eleven carried to
the next column, the first one of _l_-curves. But since the curves are
most advantageously added downward, it is best, when the first column
is finished, simply to remember the remainder from it, and not to set
down anything until the bottom is reached in the addition of the
second column, when the remainders, if any, from both columns can be
set down together. In this case, starting with the eleven carried and
counting the number of the _l_-curves, we find ourselves at the bottom
with twenty-four--twelve to carry, and nothing to set down except the
degree mark from the first column. With the twelve we go up the
adjoining loop column, and the sum must be even, as this place is
vacant in the answer; the _r_-curve column next, downward, and then
another row of degree marks. The succession must be obvious by this
time. When the last column, the one in loops to the extreme left, is
added, the sum has to be reduced to unity by successive halvings. Here
we seem to have eleven; hence we enter one loop, and carry five to the
next place, which, it must be remembered, is of _r_-curves. Halving
five we express the remainder by entering one of these curves, and
carry the quotient, two, to the degree mark place. Halving again gives
one in the next place, that of _l_-curves; and the work is complete.
It is recommended that this work be gone over several times for
practice, until the appearance and order of the characters and the
details of the method become familiar; that, when the work can be done
mechanically and without hesitation, the time occupied in a complete
addition of the example, and the mistakes made in it, be carefully
noted; that this be done several times, with an interval of some days
between the trials, and the result of each trial kept separate; that
the time and mistakes by the ordinary figures in the same example, in
several trials, be observed for comparison. Please pay particular
attention to the difference in the kind of work required by the two
methods in its bearing on two questions--which of them would be easier
to work by for hours together, supposing both equally well learned?
and in which of them could a reasonable degree of skill be more
readily acquired by a beginner? The answer to these questions, if the
comparison be a fair one, is as little to be doubted as is their high
importance.
_Example in addition by two notations_
77,823,876
14,348,907
8,654,912
5,764,801
4,635,857
1,594,323
6,417,728
4,782,969
83,886,075
34,012,224
2,903,111
48,828,125
1,724,826
7,529,536
43,344,817
10,000,000
8,334,712
1,953,125
11,308,417
759,375
21,180,840
9,765,625
18,643,788
1,000,000
44,739,243
1,889,568
2,517,471
40,353,607
4,438,414
1,679,616
23,708,715
11,890,625
945,754
823,543
15,308,805
60,466,176
30,685,377
10,077,696
19,416,381
43,046,721
===========
740,685,681
[Illustration]
Eight volunteer observers to whom this example has already been
submitted showed wide difference in arithmetical skill. One of them
took but a few seconds over two minutes, in the best of six trials, to
add by the usual figures, and set down the sum, but one figure in all
the six additions being wrong; another added once in ten minutes
fifty-seven seconds, and once in eleven minutes seven seconds, with
half the figures wrong each time. The last-mentioned observer had had
very little training in arithmetical work, but perhaps that gave a
fairer comparison. In the binary figures she made three additions in
between seven and eight minutes, with but one place wrong in the
three. With four of the observers the binary notation required nearly
double the time. These observers were all well practiced in
computation. Their best record, five minutes eighteen seconds, was
made by one whose best record was two minutes forty seconds in
ordinary figures. The author's own best results were two minutes
thirty-eight seconds binary, and three minutes twenty-three seconds
usual. He thus proved himself inferior to the last observer, as an
adder, by a system in which both were equally well trained; but a
greater familiarity (extending over a few weeks instead of a few
hours) with methods in binary addition enabled him to work twice as
fast with them. Of the author's nine additions by the usual figures,
four were wrong in one figure each; of his thirty-two additions by
different forms of binary notation, five were wrong, one of them in
two places. One observer found that he required one minute
thirty-three seconds to add a single column (average of five tried) by
the usual figures, and fifteen seconds to count the characters in one
(average of six tried) by the binary. Though these additions were
rather slow, the results are interesting. They show, making allowance
for the greater number of columns (three and a third times as many)
required by the binary plan, a saving of nearly half; but they also
illustrate the necessity of practice. This observer succeeded with the
binary arithmetic by avoiding the sources of delay that particularly
embarrass the beginner, by contenting himself with counting only, and
not stopping to divide by two, to set down an unfamiliar character, or
to recognize the mark by which he must distinguish his next column.
One well-known member of the Washington Philosophical Society and of
the American Association for the Advancement of Science, who declined
the actual trial as too severe a task, estimated his probable time
with ordinary figures at twenty minutes, with strong chances of a
wrong result, after all.
These statistics prove the existence of a class of persons who can do
faster and more reliable work by the binary reckoning. But too much
should not be made of them. Let them serve as specimens of facts of
which a great many more are to be desired, bearing on a question of
grave importance. Is it not worth our while to know, if we can, by
impartial tests, whether the tax imposed on our working brains by the
system of arithmetic in daily use is the necessary price of a blessing
enjoyed, or an oppression? If the strain produced by greater
complexity and intensity of mental labor is compensated by a
correspondingly greater rapidity in dealing with figures, the former
may be the case. If, on the contrary, a little practice suffices to
turn the balance of rapidity, for all but a small body of highly
drilled experts, in favor of an easier system, the latter must be.
This is the question that the readers of _Science_ are invited to help
in deciding. The difficulties attending a complete revolution in the
prevalent system of reckoning are confessedly stupendous; but they do
not render undesirable the knowledge that experiment alone can give,
whether or not the cost of that system is unreasonably high; nor
should they prevent those who accord them the fullest recognition from
assisting to furnish the necessary facts.
Those who are willing to undertake the addition on the plan proposed
or on any better plan, or who will submit it to such acquaintances,
skilled or unskilled, as may be persuaded to take the trouble to learn
the mechanism of binary adding, will confer a great favor by informing
the writer of the time occupied, and number of mistakes made, in each
addition. All observations and suggestions relating to the subject
will be most gratefully received.
Henry Farquhar.
Office of U.S. Coast Survey, Washington, D.C.
* * * * *
A catalogue, containing brief notices of many important scientific
papers heretofore published in the SUPPLEMENT, may be had gratis at
this office.
* * * * *