As we have explained, things that are distinguished are not really alike but only for certain turns taken to be so. This assimilation of things is of various grades. In arithmetic, meaning arithmetic in its most general sense, the only logical comprehension that the various numbers possess is respectively their greater or less partitionability; _m_ is the same as _n_ means in arithmetic that whatever has the numerical rank of _m_ has also precisely the numerical rank of _n_ no matter what summations or other numerical operations _m_ or _n_ may represent. Identity of this sort is arithmetical equality. It seems a simple relation for the reason that its intervention very decisively simplifies our arithmetical comprehensions. It is however a coincidence of two relations that are converse to one another. These relations are “not less than” and “not greater than.” It is universally admitted that the more inclusive a notion or concept is in extension, the more simple and primary it is than any other notion or concept included as an instance under it. Now all equality is “not less than” but not all “not less than” is necessarily equality; hence, “not less than” is a wider and more primary notion than equality. On the same considerations “not more than” is in the same case. Equality is the limiting case between the variable and logically more simple cases of “not less than” and “not more than.” The notion of quantity emerges on comparison however vague between any two objects that have size, independently of the notion of equality. If this were not true how could we have the notions of infinitely large and infinitely small. It is indeed true that without the notion of equality the theory of numbers and the mathematical analysis could subsist in a rudimentary state only, but to say that they would not exist at all is rash and not maintainable. The relations “not less than” “not more than” would still allow of some truly mathematical propositions, operations, and calculations. In that essentially qualitative notation that is ordinary language the relation that corresponds to equality is of very limited range but a relation that is analogous to “not less than,” viz., supersumption, is very efficient. With a theory of numbers and a mathematical analysis using only the relations “not less than” “not more than” in lieu of the relation of equality the fundamental operations, addition and substitution, would find some scope of application and hence the derivative operations, multiplication, powering, etc., and their inversions, subtraction, division, etc., would obtain in some fashion and to some extent. This can readily be seen by any one who is familiar with the way in which expressions of inequality are used in modern mathematical analysis. FRANCIS C. RUSSELL. OBSERVATIONS ON SOME POINTS IN JAMES’S PSYCHOLOGY. II. EMOTION.