The lower line contains all the combinations which are inconsistent with either premise; we have carried out in a mechanical manner that exclusion of self-contradictories which was formerly done upon the slate or paper. Accordingly, from the remaining combinations in the upper line we can draw any inference which the premises yield. If we raise the A's we find only one, and that is C, so that A must be C. If we select the c's we again find only one which is a and also b, so that we prove that not-C is not-A and not-B.

When a disjunctive proposition occurs among the premises the requisite movements become rather more complicated. Take the disjunctive argument

A is either B or C or D,

A is not C and not D,

Therefore A is B.

As there are four terms we choose the series of sixteen combinations and place them on the highest ledge of the board but one. We raise the a's and lower the b's. But we are not to reject all the Al's as contradictory, because by the first premise A's may be either B's or C's or D's. Accordingly out of the Ab's we must select the c's, and out of these again the d's, so that only Abed will remain to be rejected finally. Joining all the other fifteen combinations together again we raise the a's and lower the AC's, and thus reject the combinations inconsistent with (2); similarly we reject the AD's which are inconsistent with (3). It will be found that there remain in addition to all the eight combinations containing a only one containing A, namely

A Bcd,

whence it is apparent that A must be B, the true conclusion of the argument.

In my previous Essayh I have described the working of two other logical problems upon the abacus, which it would be tedious to repeat in this place.

The Logical Machine.

Although the Logical Abacus considerably reduced the labour of using the Indirect Method, it was not free from the possibility of error. I thought moreover that it would afford a conspicuous proof of the generality and power of the method if I could reduce it to a purely mechanical form. Logicians had long been accustomed to speak of Logic as an Organon or Instrument, and even Bacon, while he rejected the old syllogistic logic, had insisted, in the second aphorism of his New Instrument,' that the mind required some kind of systematic aid. In the kindred science of mathematics mechanical assistance of one kind or another had long been employed. Orreries, globes, mechanical clocks, and such like instruments, are really aids to calculation and are of considerable antiquity. The arithmetical machine of Pascal is more than two centuries old, having been constructed in 1642-45. M. Thomas of Colmar has recently manufactured an arithmetical machine on Pascal's principles which is extensively employed by engineers and others who need frequently to multiply or divide. To Babbage, however, was entirely due the merit of embodying the Calculus of Differences in a machine, which thus became capable of calculating the most complicated tables of figures. It seemed strange that in the more intricate science of quantity mechanism could be applicable, whereas in the simple science of

h 'Substitution of Similars,' pp. 56-59.

qualitative reasoning, the syllogism was only by analogy or simile called an Instrument. Swift satirically described the Professors of Laputa as in possession of a thinking machine, and in 1851 Mr. Alfred Smee actually proposed the construction of a Relational machine and a Differential machine, the first of which would be a mechanical dictionary and the second a mode of comparing ideas; but with these exceptions I have not yet met with so much as a suggestion of a reasoning machine. It may be added that Mr. Smee's designs, though highly ingenious, appear impracticable, and in any case do not attempt the performance of logical inferencei.

The Logical Abacus soon suggested the notion of a Logical Machine, which, after two unsuccessful attempts, I succeeded in constructing in a comparatively simple and effective form. The details of the Logical Machine have been fully described by the aid of plates in the Philosophical Transactions, and it would be both tedious and needless to repeat the account of the somewhat intricate movements of the machine in this place.

The general appearance of the machine is shown in a plate facing the title-page of this volume. It somewhat resembles a very small upright piano or organ, and has a keyboard containing twenty-one keys. These keys are of two kinds, sixteen of them representing the terms or letters A, a, B, b, C, c, D, d, which have so often been employed in our logical notation. When letters occur on the left-hand side of a proposition, formerly called the subject, each is represented by a key on the left-hand half of the keyboard; but when they occur on the right

i See his work called 'The Process of Thought adapted to Words and Language, together with a description of the Relational and Differential Machines.' Also 'Philosophical Transactions,' [1870] vol. 160, p. 518. k Philosophical Transactions,' [1870] vol. 160, p. 497. of the Royal Society,' vol. xviii. p. 166, Jan. 20, 1870.

P. 343.

Proceedings Nature,' vol. i.

Finis.

hand side, or as it used to be called the predicate of the proposition, the letter keys on the right-hand side of the keyboard are the proper representatives. The five other keys may be called operation keys, to distinguish them from the letter or term keys. They stand for the stops, copula, and disjunctive conjunctions of a proposition. The middle key of all is the copula, to be pressed when the verb is or the sign = is met. The extreme right-hand key is called the Full Stop, because it should be pressed when a proposition is completed, in fact in the proper place of the full stop. The extreme left-hand key is used to terminate an argument or to restore the machine to its initial condition; it is called the Finis key. The last key but one on the right and left complete the whole series, and represent the conjunction or in its unexclusive meaning, or the sign which I have employed, according as it occurs in the right or left hand side of the proposition. The whole keyboard is arranged

To work the machine it is only requisite to press the keys in succession as indicated by the letters and signs of a symbolical proposition. All the premises of an argument are supposed to be reduced to the simple notation which has been employed in the previous pages. Taking then such a simple proposition as

A = AB,

we press the keys A (subject), copula, A (predicate), B (predicate), and full stop.

Full Stop.

If there be a second premise, for instance

B = BC,

we press in like manner the keys

B (subj.), copula, B (pred.), C (pred.), full stop. The process is exactly the same however numerous the premises may be. When they are completed the operator will see indicated on the face of the machine the exact combinations of letters which are consistent with the premises according to the principles of thought.

As shown in the figure opposite the title-page, the machine exhibits in front an abecedarium of sixteen combinations, exactly like that of the abacus, except that the letters of each combination are separated by a certain interval. After the above problem has been worked upon the machine the abecedarium will present the following appearance

The operator will collect the various conclusions, as for instance that A is always C, that not-C is not-B and not-A; that not-B is not-A but either C or not-C, as in the use of the Logical Slate or Abacus.