sider an instance. What is meant by the ratio of A to B may be explained to a boy, by drawing a short line A and a long line B, telling him that A is said to bear a small ratio to B; and then, after lengthening the line A, telling him that A is now said to bear a larger ratio to B. But suppose I have to explain what is meant by saying that the ratio of A to B is equal to the ratio of C to D. This conception is much more complex: instead of two different quantities and one relation, there are four different quantities and three relations. To understand the proposition, the boy has to think of A and B and their difference, and, without losing his intellectual grasp of these, he has to think of C and D and their difference, and, without losing his intellectual grasp of these, he has to think of the two differences as each having a like relation to its pair of quantities. Thus the number of terms and relations to be kept before the mind is such as to imply the cooperation of many more agents of thought, any of which being absent, the proposition cannot be understood: the boy must be older before he will understand it, and, if uncultured, will probably never understand it at all. Pass now to a conception of still greater complexity—say that the ratio of A to B varies as the ratio of C to D. Far more numerous things have now to be represented in consciousness with approximate simultaneity. A and B have to be thought of as not constant in their lengths, but as one or both of them changing in their lengths, so that their difference is indefinitely variable. Similarly with C and D. And then the variability of the ratio in each case being duly conceived in terms of lines that lengthen and shorten, the thing to be understood is, that whatever difference any change brings about between A and B, the relation it bears to one or other of them is always like that which the difference simultaneously arising between C and D bears to one or other of them. The greater multiplicity of ideas required for mentally framing this proposition evidently puts it further beyond the reach of faculties not developed by appropriate culture, or not capable of being so developed. And as the type of proposition becomes still more involved, as it does when two such groups of dependent variables are compared and conclusions drawn, it begins to require a grasp that is easy only to the disciplined mathematician.
One who does not possess that complexity of faculty which, as we here see, is requisite for the grasping of a complex conception, may, in cases like these, become conscious of his incapacity; not from perceiving what it is that he lacks, but from perceiving that, by another person, results can be achieved which he cannot achieve. But, where no such thing as the verifying of exact predictions comes in to prove to one of inferior faculty that his faculty is inferior, he is usually unaware of the inferiority. To imagine a higher mode of consciousness is in some degree to have it; so that, until he has it in some degree, he cannot really conceive of its existence. An illustration or two will make this clear.