-in every train of argumentation. Surely the constant recurrence of this explanation could not have escaped the notice of the Quarterly Reviewers; and we are surprised they did not suppress that part of their criticism upon which we have been animadverting.

But this is not all the sin which Mr. Stewart has committed in conducting his speculations on the nature and utility of axioms. He justly remarks that the axioms of geometry, and those propositions which relate to the belief in our personal identity; to our belief in the existence of the material world; and to our belief in the permanency of the laws of nature-all agree in these two important particulars: first, that from neither of these classes of truths can any direct inference be drawn for the farther enlargement of our knowledge: secondly, that, notwithstanding their barrenness in this respect, they are closely connected with every exercise of our reasoning powers. Accordingly, "I should have been inclined (says he) to comprehend, under the general title of axioms, all the truths which have been hitherto under our review, if the common usage of our language had not, in a great measure, appropriated that appellation to the axioms of mathematics; and if the view of the subject which I have taken, did not render it necessary for me to direct the attention of my readers to the wide diversity between the branches of knowledge to which they are respectively subservient."

He is not a little censured, however, for expressing this inclination, to arrange under one general title all the truths which we have been considering, and, what is worthy of notice, on account of the very reasons for which, he informs us, he declined to make the classification. If our readers will take the pains to compare the last clause of the sentence just quoted with the second paragragh on page 298, of the Quarterly Review, they will see, if we do not deceive ourselves, that the latter is merely a kind of rambling paraphrase of the former. While Mr. Stewart contents himself with barely mentioning

the fact, that there is "wide diversity" between the sciences to which the different classes of truths respectively belong, the Quarterly Reviewer can only be satisfied with a minute specification of the several particulars in which that diversity consists. The reasons, as given both by the former and by the latter, why the axioms in question should be kept separate, amount solely and simply to this-that as mathematics are conversant about hypothetical truth, and as all other sciences are conversant about contingent truth, their respective axioms ought not be confounded together.

It will be our next business to examine the justness of the Quarterly Reviewers' strictures on the speculations of Mr. Stewart concerning the object of our reasonings in mathematics; and also concerning the peculiar circumstance upon which demonstrative evidence essentially depends.-The first remark we shall make, under this head, is, that, in order to convict Mr. Stewart of absurdity, his critics have taken advantage of a somewhat ambiguous expression contained in a note, without attending to the broad and satisfactory explanation which he subsequently gives of it in the body of his work; and that, secondly, the expression alluded to is not, after all, if rightly understood, either grossly illogical, or materially erroneous in point of fact.

In a note (p. 55) Mr. Stewart has occasion to remark, that "the object of the mathematics (as will afterwards more fully appear) is not truth, but systematical connection and consistency." Again (page 123)—where the accuracy of this position was more fully to appear, and where we ought to look for the statement by which he was willing to be triedthe same doctrine is fully and formally laid down. "It was already remarked (says he) in the first chapter of this part, that whereas, in all other sciences, the propositions which we attempt to establish, express facts real or supposedin mathematics, the propositions which we demonstrate only assert a connexion between certain suppositions and certain

consequences." "Our reasonings, therefore (continues he) in mathematics, are directed to an object essentially different from what we have in view, in any other employment of our intellectual faculties; not to ascertain truths with respect to actual existences, but to trace the logical filiation of consequences which follow from an assumed hypothesis."-Now, in the first quotation, had Mr. Stewart substituted, instead of "the object of mathematics," "the object of our reasonings in mathematics," he would have avoided all the censure which has been cast upon him: for very plainly the accuracy of our reasonings in mathematics is not derived from the correctness of the principles from which they set out, and are, indeed, altogether independent on the truth or falsehood of the assumed hypothesis. Nothing, in short, is more commonly seen than the most unobjectionable accuracy in a process of reasoning, which nevertheless proceeds upon a false assumption of premises; a proceeding which was deplorably successful when adopted by Berkeley and by Hume in their argument against the existence of a material world, and which is abundantly exemplified in almost every page of the article we are now examining. Considered therefore in this point of view, the assertion of Mr. Stewart, that the object of our reasonings in mathematics "is not absolute, but hypothetical truth," cannot be invalidated.

Moreover, we believe Mr. Stewart says, with the strictest propriety, that "the object of mathematics is not truth, but systematical connexion and consistency;" and had the Quarterly Reviewers attended more critically than they seem to have done to the section in which he treats of our reasoning concerning probable or contingent truths, they would, we think, have repressed those feelings of triumph which are exhibited in their remarks on this subject. The illustration there given of the doctrine will amply repay an attentive consideration. In the meantime, that the reader may have the substance of the discussion before them, we will attempt to compress the

reasonings of Mr. Stewart into as small a compass as will be consistent with perspicuity.

When we consider how signally the conclusions obtained by our reasoning in mathematics are accustomed to fail, when applied to the practical purposes of real life, does there appear to be any thing "strangely paradoxical" in the assertion, that the object of mathematics is not absolute, but conditional truth? When, for example, we are demonstrating a property of the lever, what can be more at variance with the truth, than to abstract entirely the circumstance of its own weight, and to consider it merely as an inflexible mathematical line? Is it not, in short, a distinction as old as the science itself, that the actual state of the facts differs as much from the conclusions of mathematics, as truth differs from hypothesis— practice from theory?

It is asked whether, "when Euclid proves that the three angles of a triangle are equal to two right angles, are we to understand that this is not a "truth," but merely an instance of "systematical connexion and consistency?" Plausible as a negative answer to this question may, at first sight, appear, there is yet, in our opinion, a still more satisfactory answer in a resolute and categorical affirmative. When it can be ascertained of two given figures, one of which is a triangle, and the other a diagram containing two right angles, that the sides of each are rigorously and mathematically straight, then the demonstration in question is not, when applied to these particular figures, an instance of mere systematical consistency, but an absolute and unconditional truth. Unfortunately, however, for the Quarterly Reviewers, the imperfection of our mathematical instruments will forever prevent us from ascertaining the two facts upon the certitude of which this concession depends. We may assert without the fear of contradiction, that no diagram, which is gross enough to be an object of sense, can be said to have its sides strictly and mathematically straight; and, without involving the word, therefore, in any

"metaphysical" obscurities, we must still agree with Mr. Stewart, that the object of mathematics is not absolute, but hypothetical truth.-Some observations of the same import with the foregoing reasonings, may be found on page 165 of Mr. Stewart's work; with this slight difference only, that, for the sake of illustration, he has taken his example from the circle instead of the triangle.

The Quarterly Reviewers are further very much dissatisfied with Mr. Stewart, because he is disposed to restrict the circumstance on which demonstrative evidence depends exclusively to the province of mathematics; asserting that systematical connexion can "just as easily belong to premises, ' appealing ultimately to the immutable standards of truth and falsehood, of right and wrong,' as between any other premises whatever, however arbitrary or hypothetical;" ironically insinuating, too, as if in opposition to the speculations of Mr. Stewart, that "truth and certainty in the premises of any reasoning can at all events do no harm." Mr. Stewart will certainly smile when he reads the paragraph in the Quarterly Review from which these passages are cited, and recollects that the same conclusion, which is so anxiously attempted to be established, was long ago drawn by himself, on the very same occasion and from the very same data. "Hence it appears, (says he, p. 124) that it might be possible, by devising a set of arbitrary definitions, to form a science which, although conversant about moral, political, or physical ideas, should yet be as certain as geometry." The only desideratum is to clear away those ambiguities which at present hover around the terms belonging to moral, political, or physical subjects; and to institute definitions which, whether accordant with the facts or not, may yet render these sciences capable of systematic connexion and consistency.

We have at present but little more to say concerning the criticism before us. It may be expected, perhaps, that we should pay some attention to the observations of the Quarterly