erate hardness, and a certain smell"; then, though "this object has all these Marks," I cannot be sure that "it is an apple." It It may be only a wax counterfeit, and the deception would instantly be detected by the taste, which quality was not included in the enumeration. The Reasoning is still valid in Form, but the Major Premise is false; it covers up the Fallacy ficto universalitatis. In order to be sure that an object is properly ranked under a given class, we must be certain that it contains all the original and essential qualities of the objects denoted by the class-name ; and this certainty, in the case of real things, is unattain-. able. In our conception, we may arbitrarily restrict the meaning of the word apple, so as to exclude the quality of taste; and in this sense, the wax counterfeit is properly called an apple. But in speaking to others, the word would be understood to signify all the qualities possessed by the real things, viz. this sort of fruit; and in this meaning, the wax substitute is not an apple.

We can now see why the Reasonings of the mathematician are Demonstrative, while those of the zoologist, the botanist, and other naturalists who deal only with real things, are merely Probable or contingent. The Form is always the same; Reasoning, as such, must always be Syllogistic; and when the rules of Pure Logic are duly observed, the Consequence, or the mere deduction of the Conclusion from the Premises, must be absolutely certain. The difference, then, concerns the Premises only, the truth of which, as we have seen, is not guaranteed by the principles of Logic. The universal rule, that the Middle Term must always be distributed, requires that the predesignation all, or none, should appear in at least one of the Premises. Now, our knowledge of real things is derived solely from experience; and experience, as has been mentioned, must be restricted, from its very nature, to a limited number of examples. In respect to real ob

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jects and events, it can never extend either to the inclusion or the exclusion of all; it can never pronounce with certainty either upon all, or none. Only with reference to a certain class arbitrarily formed by the Understanding,to the very things which I am now thinking of, or which I have actually observed, and to none others, to the things which are included under this Definition, and to these only, can the finite understanding of man, so far as it is enlightened only by experience, safely pronounce upon all or none. Without such limitations, naturalists, and all others who seek to educe Science from mere experience, can never speak of all or none, without falling into the Fallacy fictie universalitatis.

The mathematician deals only with certain Concepts of Quantity, whether continuous or discrete, which are precisely limited and determined by the Definitions that he employs. The propositions which he establishes do not concern circular objects and triangular objects, which are real things, but circles and triangles, which are imaginary things as conceived by the Understanding, and which are restricted by their Definitions to the possession of those qualities only which Thought voluntarily attributes to them. Hence, the conclusions which the mathematician forms respecting them are not liable to be vitiated by the intrusion of any unexpected and counteracting elements. Any theorem, therefore, which is proved of one, must hold good of all; any property which cannot belong to one, can be possessed by none, of the class thus defined. The same measure of certainty which the student of nature obtains by Intuition respecting a single real object, the mathematician acquires respecting a whole class of imaginary objects, because the latter has the assurance, which the former can never attain, that the single object, which he is contemplating in Thought, is a perfect representative of its whole class; he has this assurance, because the whole class

exists only in Thought, and are therefore all actually before him, or present to consciousness. For example; this bit of iron, I find by direct observation, melts at a certain temperature; but it may well happen that another piece of iron, quite similar to it in external appearance, may be fusible only at a much higher temperature, owing to the unsuspected presence with it of a little more, or a little less, carbon in composition. But if the angles at the basis of this triangle are equal to each other, I know that a corresponding equality must exist in the case of every other figure which conforms to the Definition of an isosceles triangle; for that Definition excludes every disturbing element. The conclusion in this latter case, then, is Universal, while in the former, it can be only Singular or Particular.

Conclusions which are demonstratively certain and absolutely universal can be obtained only when we are reasoning about abstract conceptions. In the case of natural objects and events, which can be known only through experience, we approximate universality and certainty in reasoning only by the aid of Induction and Analogy. The lack of certainty is a consequence of the lack of universality. No doubt affects the few instances which I am now actually observing, or which are present to sense or consciousness. Of these, I am as certain as of any conclusions in arithmetic or geometry. The doubt comes in only when I attempt to extend the conclusion from some, which I have examined, to all others, of which I know nothing, except from testimony, Induction, or Analogy. And this doubt is inevitable; no matter how many cases have been examined, experience can never extend to all. The fact that all matter gravitates, or has weight, is a truth which rests upon as large a testimony from experience as has ever been collected. Yet the chemist will readily admit that it is not only conceivable, but we may almost say probable,

that some of the imponderable agents, as they are called,heat, light, electricity, &c.,- may at last be found to be material; and the astronomer has not yet proved entirely to his satisfaction, that the law of gravitation is universal throughout the stellar system. From the nature of the case, he would say, the fact does not admit of absolute proof.

It appears, then, that the range of Deductive reasoning and Demonstrative proof is not confined to pure Mathematics. Whenever the objects about which we reason are pure Concepts, or mere creations of the intellect, strictly limited by Definition, and thus guarded against reference to things actually existent in Nature, our conclusions respecting them, if obtained in strict uniformity with logical rules, are as absolute as the truths of the multiplicationtable. But Mathematics, it must be admitted, afford vastly the larger number of conclusions of this class; in no other science is Demonstrative reasoning either carried so far, or so fruitful in results. This peculiarity seems to be due to the nature of those Concepts, quantity, space, and number, with which the mathematician deals. Two of these, quantity and number, are universal attributes, as they belong to all things, both to objects of sense and consciousness; and the third, space or extension, is an attribute of all external things. They are suggested to us on a greater variety of occasions than any other qualities, and thus are more frequent objects of contemplation, and more fully determined. Propositions concerning numbers," as Mr. Mill observes, "have this remarkable peculiarity, that they are propositions concerning all things whatever, — all objects, all existences of every kind, known to our experience. All things possess quantity; consist of parts which can be numbered; and, in that character, possess all the properties which are called properties of numbers."

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Again, the various modes, properties, and relations of quantity, space, and number admit of being more accurately

defined and clearly determined than those of any other class of ideas; they are separable from each other by lines of demarcation that cannot be overlooked or mistaken. Differences of degree, with which we are chiefly concerned in the case of all other qualities, are not by any means so definite, as they are shaded into each other by imperceptible gradations; their minute differences are inappreciable either by the senses or by the understanding. But the difference between two quantities, whether of number or extension, may be reduced as low as we please, and still remain as distinct to our apprehension as if it were worldwide.

But the chief peculiarity of these three Concepts, which causes them to afford so broad and fruitful a field for Demonstrative reasoning, is the measureless variety of accurately determinable relations in which all their modes stand to each other. Any one quantity stands in a perfectly conceivable ratio whether it can be exactly expressed in numbers or not to every other quantity, and also has a countless number of peculiar relations in which it stands to many at once. Attempt to enumerate, for instance, the properties of the number 9;- that it is the square of 3, the square-root of 81, the double of 41, the half of 18, &c.,

and we soon abandon the undertaking in despair. And when we come to think of the relations of these relations, as in the doctrine of proportions, it becomes evident that the properties of quantity are too great to be numbered. The field of investigation is infinite.

These innumerable and perfectly definite relations, which subsist between distinct quantities, furnish an inexhaustible number of Middle Terms, through which we obtain, by Mediate Inference, such Conclusions as are not apparent at a glance, or by direct Intuition. When the geometer, for instance, cannot determine directly the distance from one point to another, he constructs a triangle, the base of which,