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 connexion 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 existencies, 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 connexion 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, al

* 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 conditional 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.

though 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 connexion 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 hypothesis concerning 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 rea

soning 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 im mutable 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 geometricians there exists nothing which, in point of force and of subtilty, can be compared to the works of the Roman law. yers. 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 reason 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 particular 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 jurisprudence anywhere 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 consequence of new statutes, or of ancient traditions, the conclusions which the assumed hypothesis enders it necessary to in

corporate with the eternal dictates of right reason, are deduced with the soundest logic, and with an ingenuity which excites admiration. Nor are these deviations from the law of nature so frequent as is commonly imagined."*

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 resting ultimately on the evidence of our imperfect senses. The accuracy of his reasoning could never bestow on his proposition that peculiar evidence which is properly called mathematical, as long as the fact remained uncertain, whether all

* 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 orignality, as "an enormous mass of confusion and inconsistency." Making all due allowances for the exaggerations 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.

the straight lines drawn from the centre to the circumference of the figure were mathematically equal.

These observations lead me to remark a very common misconception concerning mathematical definitions; which are of a nature essentially different from the definitions employed in any of the other sciences. It is usual for writers on logic, after taking notice of the errors to which we are liable in consequence of the ambiguity of words, to appeal to the example of mathematicians, as a proof of the infinite advantage of using, in our reasonings, such expressions only as have been carefully defined. Various remarks to this purpose occur in the writings both of Mr. Locke and of Dr. Reid. But the example of mathematicians is by no means applicable to the sciences in which these eminent philosophers propose that it should be followed; and, indeed if it were copied as a model in any other branch of human knowledge, it would lead to errors fully as dangerous as any which result from the imperfections of language. The real fact is, that it has been copied much more than it ought to have been, or than would have been attempted, if the peculiarities of mathematical evidence had been attentively considered.

That in mathematics there is no such thing as an ambiguous word, and that it is to the proper use of defini tions we are indebted for this advantage, must unquestionably be granted. But this is an advantage easily secured, in consequence of the very limited vocabulary of mathematicians, and the distinctness of the ideas about which their reasonings are employed. The difference, besides, in this respect, between mathematics and the other sciences, however great, is yet only a difference in degree; and is