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 out of the A’s, which remain, we lower the *b*’s. But we are not to reject all the A*b*’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 A*b*’s we must select the *c*’s, and out of these again the *d*’s, so that only A*bcd* will remain to be rejected finally. Joining all the other fifteen combinations together again, and proceeding to premise (2), 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 AB*cd*, whence it is apparent that A must be B, the ordinary conclusion of the argument. In my “Substitution of Similars” (pp. 56–59) I have described the working upon the Abacus of two other logical problems, 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 Lord 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 Abacus is still in common use in Russia and China. The calculating machine of Pascal is more than two centuries old, having been constructed in 1642–45. M. Thomas of Colmar manufactures an arithmetical machine on Pascal’s principles which is employed by engineers and others who need frequently to multiply or divide. To Babbage and Scheutz is 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 should be applicable, whereas in the simple science of qualitative reasoning, the syllogism was only called an instrument by a figure of speech. It is true that 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