we may then substitute for B in either proposition its equivalent in the other, getting in both cases A = BC; in this we may if we like make a second substitution for B, getting A = AC. Thus, since “The Mont Blanc is the highest mountain in Europe, and the Mont Blanc is deeply covered with snow,” we infer by an obvious substitution that “The highest mountain in Europe is deeply covered with snow.” These propositions when rigorously stated fall into the forms above exhibited. This mode of inference is constantly employed when for a term we substitute its definition, or *vice versâ*. The very purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase. Thus, when we say “A circle is a curve of the second degree,” we may substitute a definition of the circle, getting “A curve, all points of which are at equal distances from one point, is a curve of the second degree.” The real forms of the propositions here given are exactly those shown in the symbolic statement, but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity. In scientific treatises a term and its definition are often both given in the same sentence, as in “The weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its mass.” The conjunction *or* in this statement gives the force of equivalence to the parenthetic phrase, so that the propositions really are Weight of a body = force with which the earth attracts it. Weight of a body = weight, &c. proportional to its mass. A slightly different case of inference consists in substituting in a proposition of the form A = AB, a definition of the term B. Thus from A = AB and B = C we get A = AC. For instance, we may say that “Metals are elements” and “Elements are incapable of decomposition.” Metal = metal element. Element = what is incapable of decomposition. Hence Metal = metal incapable of decomposition. It is almost needless to point out that the form of these arguments does not suffer any real modification if some of the terms happen to be negative; indeed in the last example “incapable of decomposition” may be treated as a negative term. Taking A = metal B = element C = capable of decomposition *c* = incapable of decomposition; the propositions are of the forms A = AB B = *c* whence, by substitution, A = A*c*. *Inference of a Partial from Two Partial Identities.* However common be the cases of inference already noticed, there is a form occurring almost more frequently, and which deserves much attention, because it occupied a prominent place in the ancient syllogistic system. That system strangely overlooked all the kinds of argument we have as yet considered, and selected, as the type of all reasoning, one which employs two partial identities as premises. Thus from the propositions