the syllogistic theory, that, in all the sciences, the words which we employ have, in the course of our previous studies, been brought to a sense as unequivocal as the phraseology of mathematicians. They proceed on the supposition, therefore, that by far the most difficult part of the logical problem has been already solved. Should the period ever arrive when the language of moralists and politicians shall be rendered as perfect as that of geometers and algebraists, then, indeed, may such contrivances as the Ars Combinatoria and the Alphabet of Human Thoughts become interesting subjects of philosophical discussion; although the probability is, that, even were that era to take place, they would be found nearly as useless in morals and politics as the syllogistic art is acknowledged to be at present, in the investigations of pure geometry. of the peculiar and supereminent advantage possessed by mathematicians, in consequence of those fixed and definite relations which form the objects of their science, and the correspondent precision in their language and reasonings, I can think of no illustration more striking than what is afforded by Dr. Halley's Latin version from an Arabic manuscript, of the two books of Apollonius Pergæus de Sectione Rationis. The extraordinary circumstances under which this version was attempted and completed (which I presume are little known beyond the narrow circle of mathematical readers) appear to me so highly curious, considered as matter of literary history, that I shall copy a short detail of them from Halley's preface. After mentioning the accidental discovery in the Bodleian library by Dr. Bernard, Savilian Professor of astronomy, of the Arabic version of Apollonius, nepi 20you anotours, Dr. Halley proceeds thus : Delighted, therefore, with the discovery of such a treasure, Bernard applied himself diligently to the task of a Latin translation. But before he had finished a tenth part of his undertaking, he abandoned it altogether, either from his experience of its growing difficulties, or from the pressure of other avocations. Afterwards, when, on the death of Dr. Wallis, the Savilian professorship was bestowed on me, I was seized with a strong desire of making a trial to complete what Bernard had begun ;-an attempt of the boldness of which the reader may judge, when he is informed, that, in addition to my own entire ignorance of the Arabic language, I had to contend with the obscurities occasioned by innumerable passages which were either defaced or altogether obliterated. With ihe assistance, however, of the sheets which Bernard had left, and which served me as a key for investigating the sense of the original, I began first with making a list of those words, the signification of which his version had clearly ascertained; and then proceeded, by comparing these words, wherever they occurred, with the train of reasoning in which they were involved, to decypher, by slow degrees, the import of the context; till at last I succeeded in mastering the whole work, and in bringing my translation (without the aid of any other person to the form in which I now give it to the public." (Appollon. Perg. de Sectione Rationis, &c. Opera et Studio Edm. Halley. Oxon. 1706. In Præfat.) When a similar attempt shall be made with equal success, in decyphering a moral or a political treatise written in an unknown tongue, then, and not till then, may we think of comparing the phraseology of these two sciences with the simple and rigorous language of the Greek geometers; or with the more refined and abstract, but not less scrupulously logical system of signs, employed by modern mathematicians. It must not, however, be imagined, that it is solely by the nature of the ideas which form the objects of its reasonings, even when combined with the precision and unambiguity of its phraseology, that mathematics is distinguished from the other branches of our knowledge. The truths about which it is conversant, are of an order altogether peculiar and singular; and the evidence of which they admit resembles nothing, either in degree or in kind, to which the same name is given, in any of our other intellectual pursuits. On these points, also, Leibnitz and many other great men bave adopted very incorrect opinions; and, by the authority of their names, have given currency to some logical errors of fundamental importance. My reasons for so thinking I shall state as clearly and fully as I can, in the following section. SECTION III. OF MATHEMATICAL DEMONSTRATION. 1.–Of the Circumstance on which Demonstrative Evidence essen tially 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 endeavor 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, although 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 connexion 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 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 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 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.* * 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 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 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 hypotheses 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 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 writing of geometricians, 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 reasoning. In what respects my opinion on this subject differs from his, will appear afterwards. Essais de Philosophie, tom. ii. p. 9, et seq. 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 renders it necessary to incorporate with the external 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.”. (Leibnitz, Op. tom. iv. p. 254.) 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 deterinined by inore 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 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 * 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 acute. ness and originality, as “an enormous mass of confusion and inconsistency.” Making all due allowances for the exaggerations of Leibnitz, it is difficult to con. ceive that his opinion, on a subject which he had so profoundly studied, should be so very widely at variance with the truth. |