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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 explanation of the peculiarity already remarked in mathematical evidence.*
The prosecution of this last idea properly belongs to the subject of mathematical demonstration, of which I intend to treat afterwards. In the mean time, I trust, that enough has been said to correct those misapprehensions of the nature of axioms, which are countenanced by the speculations, and still more by the phraseology, of some late eminent writers. On this article, my own opinion coincides very nearly with that of Mr. Locke-both in the view which he has given of the nature and use of axioms in geometry, and in what he has so forcibly urged concerning the danger, in other branches of knowledge, of attempting a similar list of maxims, without a due regard to the circumstances by which different sciences are distinguished from one another. With Mr. Locke, too, I must beg leave to guard myself against the possibility of being misunderstood in the illustrations which I have offered of some of his ideas: And for this purpose, I cannot do better than borrow his words. "In all that is here suggested concerning the little use of "axioms for the improvement of knowledge, or dangerous use in "undetermined ideas, I have been far enough from saying or intend"ing they should be laid aside, as some have been too forward to "charge me. I affirm them to be truths, self-evident truths; and "so cannot be laid aside. As far as their influence will reach, it is "in vain to endeavour, nor would I attempt to abridge it. But yet, "without any injury to truth or knowledge, I may have reason to "think their use is not answerable to the great stress which seems "to be laid on them, and I may warn men not to make an ill use of "them, for the confirming themselves in errour."†
considered as a sort of reproach to mathematicians; and I have little doubt that this reproach will continue to exist, till the basis of the science be somewhat enlarged, by the introduction of one or two new definitions, to serve as additional principles of geometrical reasoning.
For some farther remarks on Euclid's Axioms, see note (A.)
The edition of Euclid to which I uniformly refer, is that of David Gregory. Oxon. 1713.
⚫ D'Alembert, although he sometimes seems to speak a different language, approached nearly to this view of the subject when he wrote the following passage:
"Finally, it is not without reason that mathematicians consider definitions as principles; since it is on clear and precise definitions that our knowledge rests in those sciences, where our reasoning powers have the widest field opened for their exercise.”—“ Au reste, ce n'est pas sans raison que les mathématiciens regardent les définitions comme des principes, puisque, dans les sciences ou le raisonnement a la meilleure part, c'est sur des définitions nettes et exactes que nos connoissances sont appuyêes."—Elemens de Phil. p. 4.
+ Locke's Essay, Book IV. ch. vii. § 14,
After what has been just stated, it is scarcely necessary for me again to repeat, with regard to mathematical axioms, although they are not the principles of our reasoning, either in arithmetic or in hin truth geometry, their truth is supposed or implied in all our reasonings in both; and, if it were called in question, our further 's supposed be impossible. In both of these respects, we shall find them analo
gous to the other classes of primary or elementary truths which remain to be considered.
Nor let it be imagined, from this concession, that the dispute turns
The disputemerely on the meaning annexed to the word principle. It turns
not about upon an important question of fact; Whether the theorems of geometry rest on the axioms, in the same sense in which they rest on 7 word the definitions? or (to state the question in a manner still more obprinciple vious,) 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 axiom to the laws of motion, and
• "Mathematics, once fairly established on the foundation of a few axioms und 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 edi
"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 regulae 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. Philosophiae Naturalis Principia Mathematica. At the beginning, too, of Newton's Optics, the title of axioms is given to the following propositions:
" AXIOM 1.
"The angles of reflection and refraction lie in one and the same plane with the angle of incidence.
"The angle of reflection is equal
to the angle of incidence.
"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.
"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 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.
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, as the prevailing mistake, with respect to the nature of mathematical axioms, has contributed much to the support of a very erroneous theory concerning mathematical evi-math. dence, 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 cha- resolvobi racteristical cogency of mathematical demonstration. into iden
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.*
Continuation of the same Subject.
THE difference of opinion between Locke and Reid, of which 1 took notice in the foregoing part of this section, appears greater
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. 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 identicae, et repraesentantur hac formula, a =a." This sentence, which I quote from a dissertation published at Berlin about fifty years ago, express es, 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. +[All the propositions of mathematicians are identical, and are represented by this formula, a =a.]
than it really is, in consequence of an ambiguity in the word principle, as employed by the latter. In its proper acceptation, it seems Definition to me to denote an assumption (whether resting on fact or on hyDefinition a train of reasoning proceeds;
of term and for the falsity or incorrectness of which no logical rigour in the principle 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.
In cot sense may max
is called Ipin ciple?
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 commences.
Dr. Reid often uses the word in this sense, as, for example, in the Principle Triple following sentence, already quoted: "From three or four axioms, "which he calls regulae philosophandi, together with the phenomena in wt. Sense 66 observed by the senses, which he likewise lays down as first principles, und by Jr. "Newton deduces, by strict reasoning, the propositions contained in Mird?
"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 revolves 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 now been defined; and that accuracy requires, that the word, on which the whole question hinges, should not be used in both sen
ses, 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 of reasonin 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 necessa"ry 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 <6 every step of scientifical or demonstrative reasoning, gave occasion, "I imagine, to that mistaken axiom, that all reasoning was ex prae"cognitis et praeconcessis, which how far it is mistaken I shall have "occasion to show more at large, when I come to consider proposi❝tions, and particularly those propositions which are called maxims, "and to show that it is by a mistake that they are supposed to be "the foundation of all our knowledge and reasonings."
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 be"fore 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.
Dif: bet: principle