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 laws of motion,† and also to those general experimental truths which form the groundwork 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

"Mathematics once fairly established on the foundation of a few axioms and definitions, 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 immovable on the basis of self-evident principles."-Ibid. See also pp. 647, 648.

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

"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 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.

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 favor of this theory, I sball afterwards have occasion to take notice. present, it is sufficient for me to remark, (and this, I flatter myself I may venture to do with some confidence, after the foregoing rea sonings,) that in so far as it rests on the supposition that all geomet rical 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.

Continuation of the same Subject.

II. 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 falisity or incorrectness of which no logical rigor 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; t'e 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

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 St. 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 seience 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.

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 commences.

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 regulæ 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."

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 often been 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 that are intuitively known, in order to deduce from them such as require proof.

"But I cannot reconcile this with what he says (section 8th of the same chapter): The necessity of this intuitive knowledge in every step of scientifical or demonstrative reasoning, gave occasion, I imagine, to that mistaken axiom, that all reasoning was ex præ cognitis et præconcessis, which how far it is mistaken I shall have occasion to show more at large when I come to consider propositions, and particularly those propositions which are called maxims, and to show that it is by a mistake that they are supposed to be the fourdation of all our knowledge and reasonings."" (Essays on Int. Powers, p. 643, 4to. edit.)

The distinction which I have already made between elements of reasoning, and first principles of reasoning, appears to myself to throw much light on these apparent contradictions.

That the seeming difference of opinion on this point between these two profound writers, arose chiefly from the ambiguities of language, may be inferred from the following acknowledgment of Dr. Reid, which immediately follows the last quotation :

"I have carefully examined the chapter on maxims, which Mr. Locke here refers to, and though one would expect, from the quotation last made, that it should run contrary to what I have before delivered concerning first principles, I find only two or three sentences in it, and those chiefly incidental, to which I do not assent." (Essays on Int. Powers, p. 643, 4to. edit.)

Before dismissing this subject, I must once more repeat, that the doctrine which I have been attempting to establish, so far from degrading axioms from that rank which Dr. Reid would assign them, tends to identify them still more than he has done with the exercise of our reasoning powers; inasmuch as, instead of compar ing them with the data, on the accuracy of which that of our conclusion necessarily depends, it considers them as the vincula which give coherence to all the particular links of the chain; or, (to vary the metaphor) as component elements, without which the faculty of reasoning is inconceivable and impossible.*

* D'Alembert has defined the word principle exactly in the sense in which I have used it; and has expressed himself (at least on one occasion) nearly as I have done, on the subject of axioms. He seems, however, on this, as well as on

From what principle are the various properties of the circle derived, but from the definition of a circle? From what principle the properties of the parabola or ellipse, but from the definitions of these curves? A similar observation may be extended to all the other theorems which the mathematician demonstrates; and it is this observation (which, obvious as it may seem, does not appear to have occurred, in all its force, either to Locke, to Reid, or to Campbell,) that furnishes, if I mistake not, the true

son; Magnitudes which coincide with one another, that is which exactly fill the same space, are equal to one another." This, in truth, is not an axiom, but a definition. It is the definition of geometrical equality ;-the fundamental principle upon which the comparison of all geometrical magnitudes will be found ultimately to depend.

For some of these slight logical defects in the arrangement of Euclid's definitions and axioms, an ingenious, and, I think, a solid apology has been offered by M. Prévost, in his Essais de Philosophie. According to this author, if I rightly understand his meaning, Euclid was himself fully aware of the objections to which this part of his work is liable; but found it impossible to obviate them, without incurring the still greater inconvenience of either departing from those modes of proof which he had rosolved to employ exclusively in the composition of his Elements; or of revolting the student, at his first outset, by prolix and circui tous demonstrations of manifest and indisputable truths.-I shall distinguish by Italics, in the following quotation, the clauses to which I wish more particularly to direct the attention of my readers.

"C'est donc l'imperfection (peat-être inévitable) de nos conceptions, qui a engagé à fair entrer les axiomes pour quelque chose dans les principes des sciences de raisonnement pur. Et ils y font un double office. Les uns remplacent des definitions. Les autres remplacent des propositions susceptible d'être démontrées. J'en donnerai des exemples tirés des Elémens d'Euclide.

"Les axiomes remplacent quelquefois des definitions très faciles à faire, comme celle du mot tout. (El. Ax. 9.) D'autres suppléent à certaines définitions difficiles et qu'on évite, commes celles de la ligne droit et de l'angle.

Quelques axiomes remplacent des théorêmes. J'ignore si (dans les principes d'Euclide) l'axiome 11. peut-être démontré (comme l'ont cru Proclus et tant d'autres anciens et modernes). S'il peut l'être, cet axiome supplée à une démonstration probablement laborieuse.

"Puisque les axiomes ne font autre office que suppléer à des definitions et à des théorêmes, on demandera peut-être qu'on s'en passe. Observons 1. Qu'ils évitent souvent des longueurs inutiles. 2. Qu'ils tranchent les disputes à l'époque même où la science est imparfaite. 3. Que s'il est un état, auquel la science puisse s'en passer (ce que je n'affirme point) il est du moins sage, et même indispensable, de les employer, tant que quelque insuffisance, dans ce degré de perfection où l'on tend, interdit un ordre absolument irréprochable. Ajoutons 4. Que dans chaque science il y a ordinairement un principe qu'on pourroit appeller dominant, et qui par cette raison seule (et indépendamment de celles que je viens d'alléguer) a paru devoir être sorti, pour ainsi dire, du champ des définitions pour être mis en vue sous forme d'axiome. Tel me paroit être en géométrie le principe de congruence contenu dans le 8 Axiome d'Euclide." (Essais de Philosophie, tom. ii. pp. 30-32.) These remarks go far, in my opinion, towards a justification of Euclid for the Jatitude with which he has used the word axiom in his Elements. As in treating, however, of the fundamental laws of human belief, the utmost possible precision of language is indispensably necessary, I must beg leave once more to remind my readers, that, in denying Axioms to be the first principles of reasoning in mathematics, I restrict the meaning of that word to such as are analogous to the first seven in Euclid's list. Locke, in what he has written on the subject, has plainly understood the word in the same limited sense.

The edition of Euclid to which I uniformly refer, is that of David Gregory. Oxon. 1713.

By introducing for example, the idea of motion, which he has studied to avoid, as much as possible, in delivering the Elements of Plane Geometry.