perimental truths which form the ground-work of our general reasonings in catoptrics and dioptrics. For such a misapplication of the technical terms of mathematics some apology might perhaps be made, if the author had been treating on any subject connected with moral science; but surely, in a work entitled "Mathematical Principles of Natural Philosophy," the word axiom might reasonably have been expected to be used in a sense somewhat analogous to that which every person liberally educated is accustomed to annex to it, when he is first initiated into the elements of geometry.

The question to which the preceding discussion relates is of the greater consequence, that the prevailing mistake with respect to the nature of mathematical axioms, has contributed much to the support of a very erroneous theory concerning mathematical evidence, which is, I believe, pretty generally adopted at present,—that it all resolves ultimately into the perception of identity; and that it is this circumstance which constitutes the peculiar and characteristical cogency of mathematical demonstration.

AXIOM IV.

*Refraction out of the rarer medium into the denser, is made towards the perpendicular; that is, so that the angle of refraction be less than the angle of incidence.

AXIOM V.

"The sine of incidence is either accurately, or very nearly in a given ratio to the sine of refraction."

When the word axiom is understood by one writer in the sense annexed to it by Euclid, and by his antagonist in the sense here given to it by Sir Isaac Newton, it is not surprising that there should be apparently a wide diversity between their opinions concerning the logical importance of this class of propositions.

F

Of some of the other arguments which have been alleged in favour of this theory, I shall afterwards have occasion to take notice. At present it is sufficient for me to remark, (and this I flatter myself I may venture to do with some confidence, after the foregoing reasonings,) that in so far as it rests on the supposition that geometrical truths are ultimately derived from Euclid's axioms, it proceeds on an assumption totally unfounded in fact, and indeed so obviously false, that nothing but its antiquity can account for the facility with which it continues to be admitted by the learned.*

SECTION I.

II.

Continuation of the same Subject.

THE difference of opinion between Locke and Reid, of which I took notice in the foregoing part of this section, appears greater than it really is, in consequence of an ambiguity in the word principle, as employed by the latter.

* A late mathematician of considerable ingenuity and learning, doubtful, it should seem, whether Euclid had laid a sufficiently broad foundation for mathematical science in the axioms prefixed to his Elements, has thought proper to introduce several new ones of his own invention. The first of these is, that "Every quantity is equal to itself;" to which he adds afterwards, that "A quantity expressed one way is equal to itself expressed any other way.”—See Elements of Mathematical Analysis, by Professor Vilant of Saint Andrews. We are apt to smile at the formal statement of these propositions; and yet, according to the theory alluded to in the text, it is in truths of this very description that the whole science of mathematics not only begins but ends. Omnes mathematicorum propositiones sunt identicæ, et repræsentantur hac formula, "aa." This sentence, which I

In its proper acceptation, it seems to me to denote an assumption (whether resting on fact or on hypothesis) upon which, as a datum, a train of reasoning proceeds; and for the falsity or incorrectness of which no logical rigour in the subsequent process can compensate. Thus the gravity and the elasticity of the air are principles of reasoning in our speculations about the barometer. The equality of the angles of incidence and reflection; the proportionality of the sines of incidence and refraction; are principles of reasoning in catoptrics and in dioptrics. In a sense perfectly analogous to this, the definitions of geometry (all of which are merely hypothetical) are the first principles of reasoning in the subsequent demonstrations, and the basis on which the whole fabric of the science rests.

I have called this the proper acceptation of the word, because it is that in which it is most frequently used by the best writers. It is also most agreeable to the literal meaning which its etymology suggests, expressing the original point from which our reasoning sets out or commen

ces.

for ex

Dr. Reid often uses the word in this sense, as, ample, in the following sentence, already quoted: "From three or four axioms, which he calls regulæ philosophandi, together with the phenomena observed by the senses,

quote from a dissertation, published at Berlin about fifty years ago, expresses, in a few words, what seems to be now the prevailing opinion, (more particularly on the Continent) concerning the nature of mathematical evidence. The remarks which I have to offer upon it, I delay till some other questions shall be previously considered.

which he likewise lays down as first principles, Newton deduces, by strict reasoning, the propositions contained in the third book of his Principia, and in his Optics."

On other occasions, he uses the same word to denote those elemental truths (if I may use the expression,) which are virtually taken for granted or assumed, in every step of our reasoning; and without which, although no consequences can be directly inferred from them, a train of reasoning would be impossible. Of this kind in mathematics, are the axioms, or (as Mr. Locke and others frequently call them,) the maxims; in physics, a belief of the continuance of the Laws of Nature; in all our reasonings, without exception, a belief in our own identity, and in the evidence of memory. Such truths are the last elements into which reasoning resolves itself, when subjected to a metaphysical analysis; and which no person but a metaphysician or a logician ever thinks of stating in the form of propositions, or even of expressing verbally to himself. It is to truths of this description that Locke seems in general to apply the name of maxims; and in this sense, it is unquestionably true, that no science (not even geometry) is founded on maxims as its first principles.

In one sense of the word principle, indeed, maxims may be called principles of reasoning; for the words principles and elements are sometimes used as synonymous. Nor do I take upon me to say that this mode of speaking is exceptionable. All that I assert is, that they cannot be called principles of reasoning, in the sense which has just now been defined; and that accuracy requires, that the word, on which the whole question hinges, should not be used in both senses, in the course of the same argument.

It is for this reason that I have employed the phrase principles of reasoning on the one occasion, and elements of reasoning on the other.

It is difficult to find unexceptionable language to mark distinctions so completely foreign to the ordinary purposes of speech; but, in the present instance, the line of separation is strongly and clearly drawn by this criterion,~ that from principles of reasoning consequences may be deduced; from what I have called elements of reasoning,

none ever can.

A process of logical reasoning has been often likened to a chain supporting a weight. If this similitude be adopted, the axioms or elemental truths now mentioned, may be compared to the successive concatenations which connect the different links immediately with each other; the principles of our reasoning resemble the hook, or rather the beam, from which the whole is suspended.

The foregoing observations, I am inclined to think, coincide with what was, at bottom, Mr. Locke's opinion on this subject. That he has not stated it with his usual clearness and distinctness, it is impossible to deny ; at the same time, I cannot subscribe to the following severe criticism of Dr. Reid:

"Mr. Locke has observed, That intuitive knowledge is necessary to connect all the steps of a demonstration.'

"From this, I think, it necessarily follows, that in every branch of knowledge, we must make use of truths