deal more. You state that all the properties of every sphere are derived from this generating formula; you reduce an infinitely complex system of facts to two elements; you transform sensible into abstract data; you express the essence of the sphere, that is to say, the inner and primordial cause of all its properties. Such is the nature of every true definition; it is not content with explaining a name, it is not a mere description; it does not simply indicate a distinctive property; it does not limit itself to that ticketing of an object which will cause it to be distinguished from all others. There are, besides its definition, several other ways of causing the object to be recognized; there are other properties belonging to it exclusively we might describe a sphere by saying that, of all bodies having an equal surface, it occupies the most space; or in many other ways. But such descriptions are not definitions; they lay down a characteristic and derived property, not a generating and primitive one; they do not reduce the thing to its factors, and reconstruct it before our eyes; they do not show its inner nature and its irreducible elements. A definition is a proposition which marks in an object that quality from which its others are derived, but which is not derived from others. Such a proposition is not verbal, for it teaches the quality of a thing. It is not the affirmation of an ordinary quality, for it reveals to us the quality which is the source of the rest. It is an assertion of an extraordinary kind, the most fertile and valuable of all, which sums up a whole science, and in which it is the aim of every science to be summed up. There is a definition in every science, and one for each object. We do not, in every case, possess it, but we search for it everywhere. We have arrived at defining the planetary motion by the tangential force and attraction which compose it; we can already partially define a chemical body by the notion of equivalent, and a living body by the notion of type. We are striving to transform every group of phenomena into certain laws, forces, or abstract notions. We endeavor to attain in every object the generating elements, as we do attain them in the sphere, the cylinder, the circle, the cone, and in all mathematical loci. We reduce natural bodies to two or three kinds of movement―attraction, vibration, polarization-as we reduce geometrical bodies to two or three

kinds of elements-the point, the movement, the line; and we consider our science partial or complete, provisional or definite, according as this reduction is approximate or absolute, imperfect or complete.

Section IV. The Basis of Proof in Syllogism is an Abstract Law

The same alteration is required in the Theory of Proof. According to Mill, we do not prove that Prince Albert will die by premising that all men are mortal, for that would be asserting the same thing twice over; but from the facts that John, Peter, and others, in short, all men of whom we have ever heard, have died.—I reply that the real source of our inference lies neither in the mortality of John, Peter, and company, nor in the mortality of all men, but elsewhere. We prove a fact, says Aristotle,' by showing its cause. We shall therefore prove the mortality of Prince Albert, by showing the cause which produces his death. And why will he die? Because the human body, being an unstable chemical compound, must in time be resolved; in other words, because mortality is added to the quality of man. Here is the cause and the proof. It is this abstract law which, present in nature, will cause the death of the prince, and which, being present to my mind, shows me that he will die. It is this abstract proposition which is demonstrative; it is neither the particular nor the general propositions. In fact the abstract proposition proves the others. If John, Peter, and others, are dead, it is because mortality is added to the quality of man. If all men are dead, or will die, it is still because mortality is added to the quality of man. Here, again, the part played by Abstraction has been overlooked. Mill has confounded it with Experience: he has not distinguished the proof from the materials of the proof, the abstract law from the finite or indefinite number of its applications. The applications contain the law and the proof, but are themselves neither law nor proof. The examples of Peter, John, and others, contain the cause, but they are not the cause. It is not sufficient to add up the

See the Posterior Analytics, which are much superior to the Prior-di aiviwv και προτέρων.

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cases, we must extract from them the law. It is not enough to experimentalize, we must abstract. This is the great scientific operation. Syllogism does not proceed from the particular to the particular, as Mill says, nor from the general to the particular, as the ordinary logicians teach, but from the abstract to the concrete; that is to say, from cause to effect. It is on this ground that it forms part of science, the links of which it makes and marks out; it connects principles with effects; it brings together definitions and phenomena. It diffuses through the whole range of science that Abstraction which definition has carried to its summit.

Section V.-Axioms are Relations between Abstract Truths

Abstraction explains also axioms. According to Mill, if we know that when equal magnitudes are added to equal magnitudes the wholes are equal, or that two straight lines cannot enclose a space, it is by external ocular experiment, or by an internal experiment, by the aid of imagination. Doubtless we may thus arrive at the conclusion that two straight lines cannot enclose a space, but we might recognize it also in another manner. We might represent a straight line in imagination, and we may also form a conception of it by reason. We may either study its form or its definition. We can observe it in itself, or in its generating elements. I can represent to myself a line ready drawn, but I can also resolve it into its elements. I can go back to its formation, and discover the abstract elements which produce it, as I have watched the formation of the cylinder and discover the revolution of the rectangle which generated it. It will not do to say that a straight line is the shortest from one point to another, for that is a derived property; but I may say that it is the line described by a point, tending to approach towards another point, and towards that point only: which amounts to saying that two points suffice to determine a straight line; in other words, that two straight lines, having two points in common, coincide in their entire length; from which we see that if two straight lines approach to enclose a space, they would form but one straight line, and enclose nothing at all. Here is a second method of arriving at a knowledge of the axiom, and it is

clear that it differs much from the first. In the first we verify; in the second we deduce it. In the first we find by experience that it is true; in the second we prove it to be true. In the first we admit the truth; in the second we explain it. In the first we merely remark that the contrary of the axiom is inconceivable; in the second we discover, in addition, that the contrary of the axiom is contradictory. Having given the definition of the straight line, we find that the axiom that two straight lines cannot enclose a space is comprised in it, and may be derived from it, as a consequent from a principle. In fact, it is nothing more than an identical proposition, which means that the subject contains its attribute; it does not connect two separate terms, irreducible one to the other; it unites two terms, of which the second is a part of the first. It is a simple analysis, and so are all axioms. We have only to decompose them, in order to see that they do not proceed from one object to a different one, but are concerned with one object only. We have but to resolve the notions of equality, cause, substance, time, and space into their abstracts, in order to demonstrate the axioms of equality, substance, cause, time, and space. There is but one axiom, that of identity. The others are only its applications or its consequences. When this is admitted, we at once see that the range of our mind is altered. We are no longer merely capable of relative and limited knowledge, but also of absolute and infinite knowledge; we possess in axioms facts which not only accompany one another, but one of which includes the other. If, as Mill says, they merely accompanied one another, we should be obliged to conclude with him, that perhaps this might not always be the case. We should not see the inner necessity for their connection, and should only admit it as far as our experience went; we should say that, the two facts being isolated in their nature, circumstances might arise in which they would be separate; we should affirm the truth of axioms only in reference to our world and mind. If, on the contrary, the two facts are such that the first contains the second, we should establish on this very ground the necessity of their connection; wheresoever the first may be found, it will carry the second with it, since the second is a part of it, and cannot be separated from it. Nothing can exist between them and di

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cases, we must extract from them the law. It is not enough to experimentalize, we must abstract. This is the great scientific operation. Syllogism does not proceed from the particular to the particular, as Mill says, nor from the general to the particular, as the ordinary logicians teach, but from the abstract to the concrete; that is to say, from cause to effect. It is on this ground that it forms part of science, the links of which it makes and marks out; it connects principles with effects; it brings together definitions and phenomena. It diffuses through the whole range of science that Abstraction which definition has carried to its summit.

Section V.-Axioms are Relations between Abstract Truths

Abstraction explains also axioms. According to Mill, if we know that when equal magnitudes are added to equal magnitudes the wholes are equal, or that two straight lines cannot enclose a space, it is by external ocular experiment, or by an internal experiment, by the aid of imagination. Doubtless we may thus arrive at the conclusion that two straight lines cannot enclose a space, but we might recognize it also in another manner. We might represent a straight line in imagination, and we may also form a conception of it by reason. We may either study its form or its definition. We can observe it in itself, or in its generating elements. I can represent to myself a line ready drawn, but I can also resolve it into its elements. I can go back to its formation, and discover the abstract elements which produce it, as I have watched the formation of the cylinder and discover the revolution of the rectangle which generated it. It will not do to say that a straight line is the shortest from one point to another, for that is a derived property; but I may say that it is the line described by a point, tending to approach towards another point, and towards that point only: which amounts to saying that two points suffice to determine a straight line; in other words, that two straight lines, having two points in common, coincide in their entire length; from which we see that if two straight lines approach to enclose a space, they would form but one straight line, and enclose nothing at all. Here is a second method of arriving at a knowledge of the axiom, and it is