SECTION III.

OF MATHEMATICAL DEMONSTRATION.

I.

Of the Circumstance on which Demonstrative Evidence essentially depends.

THE peculiarity of that species of evidence which is called demonstrative, and which so remarkably distinguishes our mathematical conclusions from those to which we are led in other branches of science, is a fact which must have arrested the attention of every person who possesses the slightest acquaintance with the elements of geometry. And yet I am doubtful if a satisfactory account has been hitherto given of the circumstance from which it arises, Mr. Locke tells us, that "what constitutes a demonstration "is intuitive evidence at every step ;" and I readily grant, that if in a single step such evidence should fail, the other parts of the demonstration would be of no value. It does not, however, seem to me, that it is on this consideration that the demonstrative evidence of the conclusion depends, -not even when we add to it another which is much insisted on by Dr. Reid,-that, "in demonstrative evidence, "our first principles must be intuitively certain." The inaccuracy of this remark I formerly pointed out when treating of the evidence of axioms; on which occasion I also observed, that the first principles of our reasonings in mathematics are not axioms, but definitions. It is in this last circumstance (I mean the peculiarity of reasoning from definitions) that the true theory of mathematical demonstration is to be found; and I shall accordingly endeayour to explain it at considerable length, and to state some of the more important consequences to which it leads.

That I may not, however, have the appearance of claiming, in behalf of the following discussion, an undue share of originality, it is necessary for me to remark, that the leading idea which it contains has been repeatedly started, and even to a certain length prosecuted, by different writers, ancient as well as modern; but that, in all of them, it has been so blended with collateral considerations, altogether foreign to the point in question, as to divert the attention both of writer and reader, from that single principle on which the solution of the problem hinges. The advantages which mathematics derives from the peculiar nature of those relations about which it is conversant; from its simple and definite phraseology; and from the severe logic so admirably displayed in the concatenation of its innumerable theorems, are indeed immense, and well entitled to a separate and ample illustration; but they do not appear to have any necessary connection with the subject of this section. How far I am right in this opinion, my readers will be enabled to judge by the sequel.

It was already remarked, in the first chapter of this Part, that whereas, in all other sciences, the propositions which we attempt to establish, express facts real or supposed,—in mathematics, the propositions which we demonstrate only assert a connection between certain suppositions and certain consequences. Our reasonings, therefore, 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. If from this hypothesis we reason with correctness, nothing, it is manifest, can be wanting to complete the evidence of the result; as this result only asserts a necessary connection between the supposition and the

conclusion. In the other sciences, admitting that every ambiguity of language were removed, and that every step of our deductions were rigorously accurate, our conclusions would still be attended with more or less of uncertainty; being ultimately founded on principles which may, or may not, correspond exactly with the fact.*

Hence it appears, 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. It is of no moment, whether the definitions assumed correspond with facts or not, provided they do not express impossibilities, and be not inconsistent with each other. From these principles a series of consequences may be deduced by the most unexceptionable reasoning; and the results obtained will be perfectly analogous to mathematical propositions. The terms true and false, cannot be applied to them; at least in the sense in which they are applicable to propositions relative to facts. All that can be said is, that they are or are not connected with the definitions which form the-principles of the science; and, therefore, if we choose to call our conclusions true in the one case, and false in the other, these epithets must be understood merely to refer to their connection with the data, and not to their correspondence with things actually existing, or with events which we expect to be realized in future. An example of such a science as that which I have now been describing, occurs in what has been called by some writers theoretical mechanics; in which, from arbitrary hypotheses concern

* This distinction coincides with one which has been very ingeniously illustrated by M. Prevost in his philosophical essays. See his remarks on those sciences which have for their object absolute truth, considered in contrast with those which are occupied only about condi tional or hypothetical truths. Mathematics is a science of the latter description; and is therefore called by M. Prevost a science of pure reasoning. In what respects my opinion on this subject differs from his, will appear afterwards.-Essais de Philosophie, Tom. II. p. 9. et seq.

ing physical laws, the consequences are traced which would follow, if such was really the order of nature.

In those branches of study which are conversant about moral and political propositions, the nearest approach which I can imagine to a hypothetical science, analogous to mathematics, is to be found in a code of municipal jurisprudence; or rather might be conceived to exist in such a code, if systematically carried into execution, agreeably to certain general or fundamental principles. Whether these principles should or should not be founded in justice and expediency, it is evidently possible, by reasoning from them consequentially, to create an artificial or conventional body of knowledge, more systematical, and, at the same time, more complete in all its parts, than, in the present state of our information, any science can be rendered, which ultimately appeals to the eternal and immutable standards of truth and falsehood, of right and wrong. This consideration seems to me to throw some light on the following very curious parallel which Leibnitz has drawn (with what justness I presume not to decide) between the works of the Roman civilians and those of the Greek geometers. Few writers certainly have been so fully qualified as he was to pronounce on- the characteristical merits of

both.

"I have often said, that, after the writings of geometri"cians, there exists nothing which, in point of force and "of subtilty, can be compared to the works of the Roman "lawyers. And, as it would be scarcely possible, from "mere intrinsic evidence, to distinguish a demonstration of "Euclid's from one of Archimedes or of Appollonius (the "style of all of them appearing no less uniform than if rea"son herself was speaking through their organs,) so also "the Roman lawyers all resemble each other like twin"brothers; insomuch that, from the style alone of any par

"ticular opinion or argument, hardly any conjecture could "be formed with respect to the author. Nor are the traces "of a refined and deeply meditated system of natural ju"risprudence any where to be found more visible, or in "greater abundance. And, even in those cases where its "principles are departed from, either in compliance with "the language consecrated by technical forms, or in con"sequence of new statutes, or of ancient traditions, the "conclusions which the assumed hypothesis renders it necessary to incorporate with the eternal dictates of right 66 reason, are deduced with the soundest logic, and with an "ingenuity which excites admiration. Nor are these de"viations from the law of nature so frequent as is com"monly imagined."

c

I have quoted this passage merely as an illustration of the analogy already alluded to, between the systematical unity of mathematical science, and that which is conceivable in a system of municipal law. How far this unity is exemplified in the Roman code, I leave to be determined by more competent judges.†

As something analogous to the hypothetical or conditional conclusions of mathematics may thus be fancied to take place in speculations concerning moral or political subjects, and actually does take place in theoretical mechanics; so, on the other hand, if a mathematician should affirm of a general property of the circle, that it applies to a particular figure described on paper, he would at once degrade a geometrical theorem to the level of a fact rest

* Leibnitz, Op. Tom. IV. p. 254.

+ It is not a little curious that the same code which furnished to this very learned and philosophical jurist, the subject of the eulogium quoted above, should have been lately stigmatized by an English lawyer, eminently distinguished for his acuteness and originality, as "an enormous mass of confusion and inconsistency." Making all due allowances for the exaggera tions of Leibnitz, it is difficult to conceive that his opinion, on a subject which he had so profoundly studied, should be so very widely at variance with the truth.