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rems of geometry rest on the axioms, in the same sense in which they rest on the definitions? or (to state the question in a manner still more obvious,) Whether axioms hold a place in geometry at all analogous to what is occupied in natural philosophy, by those sensible phenomena which form the basis of that science? Dr Reid compares them sometimes to the one set of propositions and sometimes to the other*. If the foregoing observations be just, they bear no analogy to either.

Into this indistinctness of language Dr Reid was probably led in part by Sir Isaac Newton, who, with a very illogical latitude in the use of words, gave the name of axioms to the law's of motion †, and also to those general experimental truths which form the ground-work of our general reasonings in catoptrics and dioptrics. For such a misapplication of the technical terms

* "Mathematics, once fairly established on the foundation of a few axioms and defi"nitions, as upon a rock, has grown from age to age, so as to become the loftiest and "the most solid fabric that human reason can boast."-Essays on Int. Powers, p. 561; 4to edition.

"Lord Bacon first delineated the only solid foundation on which natural philosophy can be built; and Sir Isaac Newton reduced the principles laid down by Bacon into "three or four axioms, which he calls regulæ philosophandi. From these, together with "the phenomena observed by the senses, which he likewise lays down as first principles, "he deduces, by strict reasoning, the propositions contained in the third book of his "Principia, and in his Optics; and by this means has raised a fabric, which is not "liable to be shaken by doubtful disputation, but stands immoveable on the basis of "self-evident principles."-Ibid. See also pp. 647, 648.

† Axiomata, sive leges Motus. Vid. Philosophic Naturalis Principia Mathematica. At the beginning, too, of Newton's Optics, the title of axioms is given to the following propositions:

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 metry.

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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

" AXIOM I.

"The angles of reflection and refraction lie in one and the same plane with the angle "of incidence.

"AXIOM II.

"The angle of reflection is equal to the angle of incidence.

"AXIOM III.

"If the refracted ray be turned directly back to the point of incidence, it shall be "refracted into the line before described by the incident ray.

"AXIOM IV.

"Refraction out of the rarer medium into the denser, is made towards the perpendi"cular; 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 Eu. clid, 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.

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.

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 all 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 *.

* 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 it"self;" 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 St Andrew's, 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, "a = a." This sentence, which I 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.

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. 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 mean

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ing which its etymology suggests, expressing the original point from which our reasoning sets out or commences.

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

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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 im possible. 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.

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