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slip and at the front is fixed, as at (9, 11, 13, 15, 17), a thick square piece of sheet lead, weighing from oz. to I oz., so adjusted that each slip will hang in stable equilibrium and in an upright position when lifted by any of the pins. The lead may be covered at the front side with white paper.
The only other requisite is a flat straight-edge or ruler of hard wood about 16 inches long, 1 inch wide, and
inch thick. It is shown in the figure at (2), and an enlarged section at (3), where the sharp edge will be seen to be strengthened with a slip of brass plate (4). It will be desirable to have a box made to hold all the slips in proper order, arranged in trays, so that any set may readily be taken out by the aid of the straight-edge, inserted under the row of top pins.
In using the abacus, one or other of the series of slips is taken out, according as the logical problem to be solved contains more or less terms. If there be only two terms, the set of four is used; if three, the set of eight; if four, the set of sixteen; and if five, the set of thirty-two slips. Thus the syllogism Barbara would require the set of eight slips (see p. 56, &c.). These must be set side by side upon the topmost ledge, as at (6): the order in which they are placed is not of any essential importance, but it is generally convenient for the sake of clearness that every positive combination should be placed on the left of the corresponding negative, and that the order shown at (7), and at pp. 52, 54, and 56, should be as much as possible maintained. When a series of the slips is resting on one of the ledges, it is evident that we may separate out those marked with A or any capital letter,
by inserting the straight edge horizontally beneath the proper row of pins, and then raising the slips and removing them to another ledge. The corresponding negative slips will be left where they were, owing to the absence of pins at the point where the straight-edge is placed. We have always the option, too, of removing either the A's or the a's, the B's or the b's, and so on. Successive movements will enable us to select any class or group out of the series thus, if we took the series of sixteen, and removed first the a's and then the b's, we should have left the class of A B's, four in number. Dr. Boole based his logical notation upon the successive selection of classes, and it is this operation of thought which is represented in a concrete manner upon the abacus.
The examples given in the text (pp. 56—59) will partly serve to illustrate the use of the abacus, but I will minutely describe one more instance. Let the premises of a problem be
Let it be required to answer any question concerning the character of the things A, B, C, D, under the above conditions.
(1) Take the set of sixteen slips and place them on the topmost or first ledge of the board.
(2) Remove the A's to the second ledge.
(3) Out of the A's, remove the B's back to the first
(4) Out of what remain, remove the C's back to the first ledge.
(5) What still remain are combinations contradicted by the first premise (a), and they are to be removed to the lowest ledge, and left there. (6) The others having been joined together again on the first ledge, remove the B's to the second ledge.
(7) Out of the B's, remove the D's to the first ledge again, and
(8) Reject to the lowest ledge the B's which are notD's, as contradictory of (ß).
(9) Similarly, in treating (y), remove the C's to the second ledge, return those which are D's, and reject the C's which are not-D's to the lowest ledge.
The combinations which have escaped rejection are all which are possible under the conditions (a) (6) and (7), and they will be found to be the following:
To obtain the description of any class, we have now only to pick out that class by the straight-edge, and observe their nature. Thus, when the A's are picked out, we find that they always bring D with them; that is to say, all the A's are D's; this being the principal result of the problem. But we may also select any other
class for examination. Thus the 's are represented by only one combination, which shows that what is not-D is neither C, B, nor A.
Even when the common conclusion of an argument is self-evident, it will be found instructive to work it upon the abacus, because the whole character of the argument and the conditions of the subject are then exhibited to the eye in the clearest manner; and while the abacus gives all conclusions which can be obtained in any other way, it often gives negative conclusions which cannot be detected or proved but by the indirect method (see p. 44). It also solves with certainty problems of such a degree of complexity that the mind could not comprehend them without some mechanical aid. In my previous little work on Pure Logic (see above, § 4o, p. 45) I have given a number of examples, the working of which may be tested on the abacus, and other examples are to be found in Dr. Boole's "Laws of Thought."