[SCIENCE.]
EXPERIMENTS IN BINARY ARITHMETIC.
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 216 (represented by a degree-mark in the fifth row of these marks, counting from the right) plus 217 + 29 (two l-curves in the fifth and third places of l-curves) plus 218 + 214 + 26 (three loops) plus 219 (the r-curve at the extreme left); while the absence of 23, 22, and 21 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
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.
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