It is to this intuition I refer the power which the mind has of discovering the relation of simple numbers. A high authority on this subject has given a somewhat different account. Dr. Whewell refers our conception of number to the sense of successiveness, or, as I would render it, the faculty which discovers the relations of Time. "The conception of number appears to require the exercise of the sense of succession. At first sight, indeed, we seem to apprehend number without any act of memory, or any reference to time; for example, we look at a horse, and see that nis legs are four, and this we seem to do at once without reckoning them. But it is not difficult to see that this seeming instantaneousness of the perception of small numbers is an illusion. This resembles the many other cases in which we perform short and easy acts so rapidly and familiarly that we are unconscious of them, as in the acts of seeing, and articulating our words. And this is the more manifest, since we begin our acquaintance with number by counting even the smallest numbers. Children, and very rude savages, must use an effort to reckon even their five fingers, and find a difficulty in going further. And persons have been known who were able by habit, or by peculiar natural aptitude, to count by dozens as rapidly as common persons can by units. We may conclude, therefore, that when we appear to catch a small number by a single glance of the eye, we do, in fact, count the units of it in a regular though very brief succession. To count requires an act of memory; of this we are sensible when we count very slowly, as when we reckon the strokes of a church clock; for in such a case we may forget in the intervals of the strokes, and miscount. Now it will not be doubted that the nature of the process in counting is the same, whether we count fast or slow There is no definite speed of reckoning at which the faculties which it requires are changed, and therefore memory, which is requisite in some cases, must be so in all." I entirely concur with this statement. I am convinced that the perception of the relations of time, is presupposed in our discerning the relations of number. But there may be more required. Dr. Whewell appends a footnote, "If any one holds number to be apprehended by a direct act of intuition, as space and time are, this view will not disturb the

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other doctrines delivered in the text." I believe that one, or unity, is involved in our primary cognition of objects. Not that I think it necessary to call in a special intuition in order to our being able to count or number; but I believe that, besides the exercise of memory, and the discovery of the relations of the succession in time, there must be the general power of discovering the relations of quantity: we must be able, not only to go over the units, but further, to discover the relations of the units and of their combinations.

To this faculty I refer all those operations in which we discover equality, or difference, or proportions of any kind, in numbers. The mental capacity is greatly aided, and its intuitive perceptions are put in a position to act more readily and extensively, through the divisions and notations by tens in our modern arithmetic ; every ten, every hundred, every thousand, and so on, comes to be regarded as a unit, and the judgments in regard to units are made to reach numbers indefinitely large. These numerical judgments admit of an application to extension in space. Fixing on a certain length, superficies or solid, as a unit, we form judgments which embrace lines or surfaces or solids never actually measured. I am persuaded that, even in its common or practical operations,―as, for example, in the measurement of distance by the eye, the mind fixes on some known and familiar length as its standard, and estimates larger space by this. Ever since Descartes conceived the method of expressing curve lines and surfaces by means of equations, mathematics may be said to be concerned with quantity as their summum genus. The judgments as intuitive are all individual, but they can be generalized, when they will assume such forms as the "Common Notions," so far as they relate to quantity, prefixed by Euclid to his Elements. "Things which are equal to the same thing are equal to one another." "If equals be added to equals, the wholes are equal." "If equals be taken from equals, the remainders are equal." "If equals be added to unequals, the wholes are unequal." "If equals be taken from unequals, the remainders are unequal." "Things which are double the same thing are equal to one another." "Things which are half the same thing are equal to one another."

1 History of Scientific Ideas, II. x. 4.

SECT. VI.-THE RELATIONS OF RESEMBLANCE.

It has been generally acknowledged that man's primary knowledge is of individual objects: not that he as yet knows them to be individual; it is only after he has been able to form general notions that he draws the distinction, and finds that what he first knew was singular. What is meant is, that the boy does not begin with a notion of man or woman, or humanity in general, but with a knowledge of a particular man, say his father, or a particular woman, say his mother; and it is only as other men and other women come under his notice, and he observes their points of agreement, that he is able to rise to the general notion of man, or woman, or humankind.

In the mental processes involved in generalization, the most important part is the observational one. When we discover, for example, the resemblance of plants, and proceed to group them into species, genera, and orders, the operation is one of induction and comparison. There is no necessity of thought involved in the law that roses have five petals, or that fishes are cold-blooded, or indeed in any of the laws of natural history. Still there are laws of thought which have a place in the generalizing process.

1. The universal implies singulars.-The mind pronounces this judgment when it looks at the nature of the individuals and the generals. The universal is not something independent of the singulars, prior to the singulars, or above the singulars. A general notion is the notion of an indefinite number of objects possessing a common attribute or attributes, and includes all the objects possessing the common quality or qualities. It is clear, therefore, that the general proceeds on and presupposes individuals. If there were no individuals, there would be no general; and if the individuals were to cease, the general would likewise cease. If there were no individual roses, there would be no such thing as a class of plants called roses.

2. When the singulars are real, the universal is also real; always, of course, on the supposition that the generalization has been properly made. There exists, we shall suppose, in nature, a number of objects possessing common attributes; we have observed

their points of resemblance, and put them in a class: has, or has not, the class an existence? In reply, I say that the genus has an existence and a reality as well as the individual objects. An indefinite number of animals chew the cud, and are called ruminant t; the class ruminant has an existence quite as much as the individual animals. But let us observe what sort of reality the class has ; it is a reality merely in the individuals, and in the possession of common qualities by these individuals.

3. Whatever is predicated of a class may be predicated of all the members of the class; and vice versa, whatever is predicated of all the members of a class, may be predicated of the class. This is a self-evident and necessary proposition. It is pronounced by the mind in an individual form whenever it contemplates the relation of a class and the members of the class; thus, if the general maxim be discovered or allowed, that all reptiles are cold-blooded, and the further fact be given or ascertained that the crocodile is a reptile, the conclusion is pronounced that the crocodile is coldblooded.

We shall discover, when we come to apply these general principles, that the laws mentioned in this section play an important part in Logic, and have a place in the Notion, in the Judgment, and in Reasoning.

SECT. VII.-RELATIONS OF ACTIVE PROPERTY.

I have been striving to prove that we cannot know either self or body acting on self, except as possessing property. On looking at the properties of objects, the mind at once pronounces certain decisions. These, like all our other intuitive judgments, have a reference, in the first instance, to the individual case presented, but may be made universal by a process of generalization. Thus, the mind declares, "this property implies a substance;" "this substance will exercise a property." The abstract truths will seldom be formally enunciated, but, as regulative principles, they underlie our common thoughts, and we proceed on them, even when entirely unaware of their nature or of their existence. Every action or manifestation we intuitively regard as the action or exhibition of

a something having a substantial being. On falling in with a new substance, say an aërolite just dropped from the heavens, we know not indeed what its properties are, but we are sure that it has properties, and we make an attempt to discover them.

SECT. VIII.-RELATION OF CAUSE AND EFFECT.

All our primitive judgments carry us back to primitive cognitions and beliefs, that is, they are pronounced by the mind as it looks to objects intuitively known or necessarily believed in. The judgment which affirms that the cause must produce its effect, and that the effect has resulted from a cause, proceeds from and is grounded on a cognition which has already passed under our notice, the intuitive knowledge of substance exercising power. It will appear, as we advance, that those who overlook or deny the mind's primary knowledge of power, can give no adequate or satisfactory account of the nature or meaning of the causal judgment.

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It will be needful to show here, first of all, that the judgment is not derived from the generalizations of outward experience. we do so, it will be necessary to state, though it will not be necessary to dwell on it after the enunciations which have been so often repeated, that our conviction is not of a general truth, but relates solely to individual facts presented to or contemplated by the mind. Our original judgment is not that every cause has an effect, and that every effect has a cause,-propositions which will not be admitted and cannot be understood till the words " cause and "effect," terms very abstract and general, be explained, but it is that this thing having power, may produce an effect, and that this thing apprehended as a new thing or as having been changed must have had a cause.

In proceeding to prove that the mental conviction, thus understood, is not derived from experience, I am disposed to admit at once that observation might, without any original intuition, lead to a loose general belief in cause and effect. On seeing two events in frequent juxtaposition, we might be disposed when we see the one to think of the other by the ordinary law of association, and when we see the one to expect the other, as the result of a process