The same double operation will be necessary whenever the proposition is not of the kind called a partial identity (p. 40). Thus AB = CD, AB = AC, A = B ꖌ C, A ꖌ B = C ꖌ D, all require to be read from both ends separately. The proper rule for using the machine may in fact be given in the following way:--(1) *Read each proposition as it stands, and play the corresponding keys*: (2) *Convert the proposition and read and play the keys again in the transposed order of the terms.* So long as this rule is observed the true result must always be obtained. There can be no mistake. But it will be found that in the case of partial identities, and some other similar forms of propositions, the transposed reading has no effect upon the combinations of the Logical Alphabet. One reading is in such cases all that is practically needful. After some experience has been gained in the use of the machine, the worker naturally saves himself the trouble of the second reading when possible. It is no doubt a remarkable fact that a simple identity cannot be impressed upon the machine except in the form of two partial identities, and this may be thought by some logicians to militate against the equational mode of representing propositions. Before leaving the subject I may remark that these mechanical devices are not likely to possess much practical utility. We do not require in common life to be constantly solving complex logical questions. Even in mathematical calculation the ordinary rules of arithmetic are generally sufficient, and a calculating machine can only be used with advantage in peculiar cases. But the machine and abacus have nevertheless two important uses. In the first place I hope that the time is not very far distant when the predominance of the ancient Aristotelian Logic will be a matter of history only, and when the teaching of logic will be placed on a footing more worthy of its supreme importance. It will then be found that the solution of logical questions is an exercise of mind at least as valuable and necessary as mathematical calculation. I believe that these mechanical devices, or something of the same kind, will then become useful for exhibiting to a class of students a clear and visible analysis of logical problems of any degree of complexity, the nature of each step being rendered plain to the eyes of the students. I often used the machine or abacus for this purpose in my class lectures while I was Professor of Logic at Owens College.